If $x - 2$ is a factor of each of the following two polynomials, find the values of $a$ in each case:$x^5 - 3x^4 - ax^3 + 3ax^2 + 2ax + 4$

Given:

Given expression is $x^5 - 3x^4 - ax^3 + 3ax^2 + 2ax + 4$.

$x - 2$ is a factor of $x^5 - 3x^4 - ax^3 + 3ax^2 + 2ax + 4$.

To do:

We have to find the value of $a$.

Solution:

We know that,

If $(x-m)$ is a root of $f(x)$ then $f(m)=0$.

Therefore,

$f(2)=0$

$\Rightarrow (2)^5-3(2)^4-a(2)^3 +3a(2)^2 + 2a(2) + 4=0$

$\Rightarrow 32-3(16)-8a+3a(4)+4a+4=0$

$\Rightarrow 32-48-8a+12a+4a+4=0$

$\Rightarrow 8a=12$

$\Rightarrow a=\frac{12}{8}=\frac{3}{2}$

The value of $a$ is $\frac{3}{2}$. 

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Updated on: 10-Oct-2022

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