Question

Factor each expression completely:

a. 9x4 − 16x2
b. 2x3 + 5x2 − 8x − 20
c. x3 + 3x2 + 3x + 1
d. 8x3 − 1

Answer 1

a.
`9x^4 - 16x^2`

`=(3x^2)^2 - (4x)^2`

By using a² - b² = ( a + b ) ( a - b ),

`= (3x^2 + 4x ) ( 3x^2 - 4x )`

`= (3x^2 + 4) ( ((√(3)x )^2 - (2√(x))^2)`

`= (3x^2 + 4) (√(3)x + 2√(x))(√(3)x - 2√(x))`

b.
`2x^3 + 5x^2 - 8x - 20`

`=x^2( 2x + 5) - 4( 2x + 5)`

`= (x^2 - 4) ( 2x + 5)`

`=(x+2)(x-2)(2x + 5)`

c.
`x^3 + 3x^2 + 3x + 1`

`=x^3 + 3x(x +1) + 1^3`

`=(x+1)^3`

( by using (a + b)³ = a³ + 3ab(a+b) + b³ )

d.
`8x^3 - 1`

`=(2x)^3 - 1`

`=(2x-1)((2x)^2 + 2x* 1 + 1^2)`

`=(2x - 1 )( 4x^2 + 2x + 1)`

( by using a³ - b³ = ( a - b ) (a² + ab + b² ) )