The polynomials (2x^3 + x^2 - ax + 2) and (2x^3 - 3x^2 - 3x + a) … - Vidyadip

Question

The polynomials$ (2x^3 + x^2 - ax + 2)$ and $(2x^3 - 3x^2 - 3x + a)$ when divided by $(x - 2)$ leave the same remainder. Find the value of a.

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Answer

Let $f(x)=2 x^3+x^2-a x+2$
$g(x)=2 x^3-3 x^2-3 x+a .$
By remainder theorem, when $f(x)$ is divided by $(x-2)$, then the remainder $=f(2)$.
Putting $x=2$ in $f(x)$, we get
$f(2)=2 \times 2^3+2^2-a \times 2+2$
$=16+4-2 a+2$
$=-2 a+22$
By remainder theorem, when $g(x)$ is divided by $(x-2)$, then the remainder $=g(2)$.
$g(2)=2 \times 2^3-3 \times 2^2-3 \times 2+a$
$=16-12-6+a$
$=-2+a$
It is given that,
$\Rightarrow-2 a+22=-2+a$
$\Rightarrow-3 a=-24$
$\Rightarrow a=8$
Thus, the value of $a$ is 8 .

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