Question

verify if -2 and 3 are zeroes of the polynomial 2x^3 -3x^2 -11x +6 if yes factorize the polynomial

Answer 1

Hey MATE!


This must be your answer.

Hope it helps #))

Answer 2

Yes, $(-2)$ and $3$ are the zeroes of the polynomial $2x^{3}-3x^{2}-11x+6$.

$2x^{3}-3x^{2}-11x+6=(x+2)(x-3)(2x-1)$

To verify $(-2)$ and $3$ are the zeroes of the given polynomial :

Let, $f(x)=2x^{3}-3x^{2}-11x+6$

When $x=-2$, we have

$\quad f(-2)=2(-2)^{3}-3(-2)^{2}-11(-2)+6$

$\Rightarrow f(-2)=-16-12+22+6$

$\Rightarrow f(-2)=0$

Since $f(-2)=0$, $(-2)$ is a zero of the given polynomial.

When $x=3$, we have

$\quad f(3)=2(3)^{3}-3(3)^{2}-11(3)+6$

$\Rightarrow f(3)=54-27-33+6$

$\Rightarrow f(3)=0$

Since $f(3)=0$, $3$ is a zero of the given polynomial.

Factorization :

Since $(-2)$ and $3$ are two zeroes $f(x)$, then certainly $(x+2)$ and $(x-3)$ are its factors, then $(x+2)(x-3)=x^{2}-x-6$ is one factor.

Now, $2x^{3}-3x^{2}-11x+6$

$=2x^{3}-2x^{2}-12x-x^{2}+x+6$

$=2x(x^{2}-x-6)-1(x^{2}-x-6)$

$=(x^{2}-x-6)(2x-1)$

$=(x+2)(x-3)(2x-1)$

• since $x^{2}-x-6=(x+2)(x-3)$

This is the required factorization.