Using factor theorem, factorize each of the following polynomials:$2x^4 - 7x^3 - 13x^2 + 63x - 45$

Given:

Given expression is $2x^4 - 7x^3 - 13x^2 + 63x - 45$.

To do:

We have to factorize the given polynomial.

Solution:

Let $f(x)=2 x^{4}-7 x^{3}-13 x^{2}+63 x-45$

The factors of the constant term $-45$ are $\pm 1, \pm 3, \pm 5, \pm 9, \pm 15$ and $\pm 45$.
Let $x=1$, this implies,

$f(1)=2(1)^{4}-7(1)^{3}-13(1)^{2}+63(1)-45$

$=2(1)-7(1)-13(1)+63(1)-45$

$=2-7-13+63-45$

$=65-65$

$=0$

Therefore, $x-1$ is a factor of $f(x)$

Let $x=3$, this implies,

$f(3)=2(3)^{4}-7(3)^{3}-13(3)^{2}+63(3)-45$

$=162-189-117+189-45$

$=351-351$

$=0$

Therefore, $x-3$ is a factor of $f(x)$

Let $x=5$, this implies,

$f(5)=2(5)^{4}-7(5)^{3}-13(5)^{2}+63(5)-45$

$=1250-875-325+315-45$

$=1565-1245$

$=320 \
eq 0$

Therefore, $x-5$ is not a factor of $f(x)$.

Let $x=-3$, this implies,

$f(-3)=2(-3)^{4}-7(-3)^{3}-13(-3)^{2}+63(-3)-45$

$=162+189-117-189-45$

$=351-351$

$=0$

Therefore, $x+3$ is a factor of $f(x)$.

Dividing $f(x)$ by $(x-1)(x-3)(x+3)=x^{3}-x^{2}-9 x+9$, we get,

$x^3-x^2-9x+9$)$2 x^{4}-7 x^{3}-13 x^{2}+63 x-45$($2x-5

                              $2x^4-2x^3-18x^2+18x$

                         ------------------------------------------

                                        $-5x^3+5x^2+45x-45$

                                        $-5x^3+5x^2+45x-45$

                           ----------------------------------------

                                                     0

Hence, $2 x^{4}-7 x^{3}-13 x^{2}+63 x-45=(2x-5)(x-1)(x-3)(x+3)$.

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Updated on: 10-Oct-2022

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