What must be subtracted from x ^ 3 - 6x ^ 2 - 15x + 80 so that the result is exactly divisible by

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By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

By divisible algorithm, when

is divided by

the reminder is a linear polynomial

Let

be subtracted from p(x) so that the result is divisible by q(x).

Let

We have,

Clearly,

and

are factors of q(x), therefore, f(x) will be divisible by q(x) if

and

are factors of f(x), i.e. f (−4) and f (3) are equal to zero.

Therefore,

and

Adding (i) and (ii), we get,

Putting this value in equation (i), we get,

Hence,

will be divisible by

if 4 x − 4 is subtracted from it

If x − 2 is a factor of each of the following two polynomials, find the values of a in each case

(i) x3 − 2ax2 + ax − 1

(ii) x5 − 3x4 − ax3 + 3ax2 + 2ax + 4

In each of the following two polynomials, find the value of a, if x − a is factor:

(i) x6 − ax5 + x4 − ax3 + 3x − a + 2

(ii x5 − a2x3 + 2x + a + 1)

Q&A What

What must be subtracted from x ^ 3 - 6x ^ 2 - 15x + 80 so that the result is exactly divisible by

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