Question

factorise the following using synthetic division method
$2x {}^{3 } - 7x {}^{2} - 10x + 24$
​

Answer 1

Solution:

Given polynomial:

$\tt\longrightarrow p(x)=2x^3-7x^2-10x+24$

If we take x = 4, we get:

$\tt\longrightarrow p(4)=2\cdot (4)^3-7\cdot(4)^2-10\cdot4+24$

$\tt\longrightarrow p(4)=128-112-40+24$

$\tt\longrightarrow p(4)=0$

Since, p(4) = 0, x - 4 is a factor of p(x) by factor theorem.

Let us divide the polynomial by x - 4.

x - 4 )   2x³ - 7x² - 10x + 24   ( 2x² + x - 6

           2x³ - 8x²

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

                      x² - 10x + 24

                      x² -   4x

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

                              -6x + 24

                              -6x + 24

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

                                    0

So, our polynomial becomes:

$\tt\longrightarrow p(x)=(2x^2+x-6)(x-4)$

$\tt\longrightarrow p(x)=(2x^2+4x-3x-6)(x-4)$

$\tt\longrightarrow p(x)=[2x(x+2)-3(x+2)](x-4)$

$\tt\longrightarrow p(x)=(2x-3)(x+2)(x-4)$

Which is the required factorization of the polynomial.

Answer:

$\tt\hookrightarrow p(x)=(2x-3)(x+2)(x-4)$