Find all the zeroes of the polynomial x^3 4x^2 - 3x+12 if two of its zeroes are sqrt 3 and sqrt 3

Answer

Verified

Hint: Use the given root as a factor and divide by combining it we need to use the quotient to calculate the third zero. If $f(x)= x+a=0$ then $x=-a$ will be a zero of $f(x)$.

Complete step by step answer:

We know that if $x=a$ is the root or zero of any polynomial $f(x)$ then $x+a$ will completely divide the given polynomial and $x+a$ will be termed a factor of that polynomial \[f(x)\].Here in question, it is given that $\sqrt{3}$ and \[-\sqrt{3}\]are the zeroes of the polynomial $f(x)={{x}^{3}}-4{{x}^{2}}-3x+12$So according to the factor theorem \[x-\sqrt{3}\]and$x+\sqrt{3}$ will be the factor of the polynomial and product of these two will completely divide the given polynomial\[f(x)\].$\left( x-\sqrt{3} \right)\left( x+\sqrt{3} \right)={{\left( x \right)}^{2}}-{{\left( \sqrt{3} \right)}^{2}}$.Using,$\left( a+b \right)(a-b)={{a}^{2}}-{{b}^{2}}$$={{x}^{2}}-3$We now divide the polynomial ${{x}^{3}}-4{{x}^{2}}-3x+12$ by ${{x}^{2}}-3$${{x}^{2}}-3\overset{x}{\overline{\left){\begin{align} & {{x}^{3}}-4{{x}^{2}}-3x+12 \\ & {{x}^{3}}-3x \end{align}}\right.}}$Here first quotient is x since first term of polynomial is${{x}^{3}}$ ,Now subtracting the term ,we get further division step as$-4{{x}^{2}}+12$since ${{x}^{3}}$ get cancelled with ${{x}^{3}}$ and $-3x$ with $-3x$So next quotient should be $-4$ to make same as $-4{{x}^{2}}+12$${{x}^{2}}-3\overset{x-4}{\overline{\left){\begin{align} & {{x}^{3}}-4{{x}^{2}}-3x+12 \\ & {{x}^{3}}-3x \\ &\hline\\ & -4{{x}^{2}}+12 \\ & -4{{x}^{2}}+12 \end{align}}\right.}}$Further subtracting the last term we can clearly see that $-4{{x}^{2}}$ get cancelled with $-4{{x}^{2}}$and $12$ with $12$ so we get remainder as zero hence \[x-4\] will be the quotient as well as the factor of the given polynomial.Therefore to calculate the third zero of the polynomial we need to put this quotient equals to zero.$\begin{align} & x-4=0 \\ & x=4 \end{align}$

Hence, third zero of the polynomial ${{x}^{3}}-4{{x}^{2}}-3x+12$ is 4.

Note:

- We can also use the sum of zeroes and product of zeroes method to find the third zero of the given polynomial.- We also know it as the relation between zeroes and coefficients of polynomials.

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