Obtain all other zeroes of (x^4 + 4x^3 − 2x^2 − 20x −15, if two of its zeroes are √5 and − √5)

Solution:

Given polynomial p(x) = 3x4 + 6x3 - 2x2 - 10x - 5

Two zeroes of the polynomial are given as √(5/3) and -√(5/3)

Therefore,

[x - √(5/3)] [x + √(5/3)] = (x2 - 5/3) is a factor of 3x4 + 6x3 - 2x2 - 10x - 5.

Therefore, we divide the given polynomial by (x2 - 5/3)

Therefore, 3x4 + 6x3 - 2x2 - 10x - 5 = (x2 - 5/3)(3x2 + 6x + 3) + 0

= 3 (x2 - 5/3) (x2 + 2x + 1)

On factorizing x2 + 2x + 1, we get (x + 1)2

Therefore, its zero is given by

x + 1 = 0

x = - 1

As it has the term (x + 1)2,

Therefore, there will be two identical zeroes at x = - 1

Hence the zeroes of the given polynomial are √(5/3) and -√(5/3), - 1 and - 1.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 2

Video Solution:

Obtain all other zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5, if two of its zeroes are √5/3 and -√5/3.

NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.3 Question 3

Summary:

All the other zeroes of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeroes are √5/3 and -√5/3 are -1, and -1.

☛ Related Questions:

• Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:(i) p(x) = x3 - 3x2 + 5x - 3, g(x) = x2 - 2(ii) p(x) = x4 - 3x2 + 4x + 5, g(x) = x2 + 1 - x(iii) p(x) = x4 - 5x + 6, g(x) = 2 - x2
• Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:(i) t2 - 3, 2t4 + 3t3 - 2t2 - 9t - 12(ii) x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2(iii) x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1
• On dividing x3 - 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and - 2x + 4, respectively. Find g (x).
• Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg-p(x) = deg q(x) (ii) deg q(x) = deg r (x) (iii) deg r (x) = 0

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Obtain all other zeros of x3+4x3 2x2 20x 15 if two of its zeros are √5 and √5.

Quotient $=x^2+4x+3=x^2+3x+x+3$

$=x(x+3)+(x+3)=(x+3)(x+1)$

Other zeros of the given polynomial are the zeros of q(x)

Therefore, $x=-3,-1$

Thus, the zeros of the given polynomial are $\sqrt{5},–\sqrt{5},-3,-1$

Secondary School Mathematics X