If $x^3 + ax^2 - bx + 10$ is divisible by $x^2 - 3x + 2$, find the values of $a$ and $b$.

Given:

Given expression is $x^3 + ax^2 - bx + 10$.

$(x^2 - 3x + 2)$ is a factor of $x^3 + ax^2 - bx + 10$.

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 $f(m)=0$.

$x^2-3x+2=x^2-2x-x+2$

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

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

This implies,

$x-2$ and $x-1$ are factors of $x^3 + ax^2 - bx + 10$.

Therefore,

$f(2)=0$

$\Rightarrow (2)^3 + a(2)^2 - b(2) + 10=0$

$\Rightarrow 8+4a-2b+10=0$

$\Rightarrow 4a-2b+18=0$

$\Rightarrow 2(2a-b+9)=0$

$\Rightarrow b=2a+9$........(i)

$f(1)=0$

$\Rightarrow (1)^3 + a(1)^2 - b(1) + 10=0$

$\Rightarrow 1+a-b+10=0$

$\Rightarrow a-b+11=0$

$\Rightarrow a-(2a+9)+11=0$........[From (i)]

$\Rightarrow a-2a-9+11=0$

$\Rightarrow -a=-2$

$\Rightarrow a=2$

$\Rightarrow b=2(2)+9$

$\Rightarrow b=4+9=13$

The values of $a$ and $b$ are $2$ and $13$ respectively.       

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Updated on: 10-Oct-2022

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