Using the factor theorem, show that (x – 2) is a factor of x3 + x2 �

1.	Using the factor theorem, show that (x – 2) is a factor of x3 + x2 – 4x – 4. Hence, factorise the polynomial completely.
Answer» Given polynomial is p(x) = x³ + x² – 4x – 4 x – 2 is its factor, if p(2) = 0 p(2) = (2)³ + (2)²– 4(2) – 4 = 8 + 4 – 8 – 4 = 0 Thus, x – 2 is a factor of p(x). Now, x³ + x² – 4x + 4 = x²(x + 1) – 4(x + 1) = (x + 1) (x² – 4) = (x + 1) (x + 2) (x – 2) Hence, the required factors are (x + 1), (x + 2) and (x – 2).

Using the factor theorem, show that (x – 2) is a factor of x3 + x2 – 4x – 4. Hence, factorise the polynomial completely.

Answer»

Given polynomial is p(x) = x³ + x² – 4x – 4

x – 2 is its factor, if p(2) = 0

p(2) = (2)³ + (2)²– 4(2) – 4 = 8 + 4 – 8 – 4 = 0

Thus, x – 2 is a factor of p(x).

Now, x³ + x² – 4x + 4 = x²(x + 1) – 4(x + 1)

= (x + 1) (x² – 4)

= (x + 1) (x + 2) (x – 2)

Hence, the required factors are (x + 1), (x + 2) and (x – 2).