Factorize each of the following polynomials:
(i) x3+13x2+31x-45 given that x+9 is a factor
(ii) 4x3+20x2+33x+18 given that 2x+3 is a factor.
(i) Let, f (x) = x3+13x2+31x-45
Given that (x + 9) is a factor of f (x)
Let us divide f (x) by (x + 9) to get the other factors
By using long division method, we have
f (x) = x3+13x2+31x-45
= (x + 9) (x2 + 4x – 5)
Now,
x2 + 4x – 5 = x2 + 5x – x – 5
= x (x + 5) – 1 (x + 5)
= (x – 1) (x + 5)
f (x) = (x + 9) (x + 5) (x – 1)
Therefore, x3+13x2+31x-45 = (x + 9) (x + 5) (x – 1)
(ii) Let, f (x) = 4x3+20x2+33x+18
Given that (2x + 3) is a factor of f (x)
Let us divide f (x) by (2x + 3) to get the other factors
By long division method, we have
4x3+20x2+33x+18 = (2x + 3) (2x2 + 7x + 6)
2x2 + 7x + 6 = 2x2 + 4x + 3x + 6
= 2x (x + 2) + 3 (x + 2)
= (2x + 3) (x + 2)
4x3+20x2+33x+18 = (2x + 3) (2x + 3) (x + 2)
= (2x + 3)2 (x + 2)
Hence,
4x3+20x2+33x+18 = (2x + 3)2 (x + 2)