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

Explanation

`45⇒±1,±3,±5,±9,±15,±45`
`x=1,p(1)=2(1)^4−7(1)^3−13(1)^2+63(1)−45`
`p(1)=2−7−13+63−45=65−65=0`
`x=1 or x - 1` is a factor of p(x).
Similarly if we put`x =3,p(3)=2(3)^4−7(3)^3−13(3)^2+63(3)−45`
`p(3)=162−189−117+189−45=162−162=0`
Hence, `x = 3 or (x - 3) = 0` is the factor of p(x).
`p(x)=2x^4−7x^3−13x^2+63x−45`
`p(x)=2x^3(x−1)−5x^2(x−1)−18x(x−1)+45(x−1)`
⇒`p(x)=(x−1)(2x^3−5x^ 2−18x+45)`
`⇒p(x)=(x−1)(2x^3−5x^2−18x+45)`
`⇒p(x)=(x−1(2x^2(x−3)+x(x−3)−15(x−3))`
`⇒p(x)=(x−1)(x−3)(2x^2+x−15)`
`⇒p(x)=(x−1)(x−3)(2x^2
+6x−5x−15)`
`⇒p(x)=(x−1)(x−3(2x(x+3)−5(x+3))`
`⇒p(x)=(x−1)(x−3)(x+3)(2x−5)`