Find the values of $a$ and $b$ so that $(x + 1)$ and $(x - 1)$ are factors of $x^4 + ax^3 - 3x^2 + 2x + b$.
Given:
Given expression is $x^4 + ax^3 - 3x^2 + 2x + b$.
$(x + 1)$ and $(x - 1)$ are factors of $x^4 + ax^3 - 3x^2 + 2x + b$.
To do:
We have to find the values of $a$ and $b$.
Solution:
We know that,
If $(x-m)$ is a root of $f(x)$ then $f(m)=0$.
Therefore,
$f(-1)=0$
$\Rightarrow (-1)^4+a(-1)^3-3(-1)^2 + 2(-1) + b=0$
$\Rightarrow 1-a-3-2+b=0$
$\Rightarrow a=b-4$...............(i)
$f(1)=0$
$\Rightarrow (1)^4+a(1)^3-3(1)^2 + 2(1) + b=0$
$\Rightarrow 1+a-3+2+b=0$
$\Rightarrow b-4+b=0$ [From (i)]
$\Rightarrow 2b=4$
$\Rightarrow b=\frac{4}{2}$
$\Rightarrow b=2$
$\Rightarrow a=2-4=-2$
The values of $a$ and $b$ are $-2$ and $2$ respectively.
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