Using factor theorem, factorize each of the following polynomial:

Question

Using factor theorem, factorize each of the following polynomial: x 3 -3 x 2 -9 x -5

Philoid · Accepted Answer

Let, f (x) = x³-3x²-9x-5

The factors of the constant term - 5 are

Putting x = -1, we have

f (-1) = (-1)³ – 3 (-1)² – 9 (-1) - 5

= -1 – 3 + 9 - 5

= 0

So, (x + 1) is a factor of f (x)

Let us now divide

f (x) = x³-3x²-9x-5 by (x + 1) to get the other factors of f (x)

Using long division method, we get

x³-3x²-9x-5 = (x + 1) (x² - 4x 5)

x² - 4x - 5 = x² - 5x + x - 5

= x (x - 5) + 1 (x - 5)

= (x + 1) (x - 5)

Hence, x³+13x²+32x+20 = (x + 1) (x + 1) (x - 5)

= (x + 1)² (x – 5)