Factorize each of the following polynomials:(i) x3 + 13x2 + 31x - 45

1.	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.
Answer» (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)

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.

Answer»

(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)