Question

Factor each expression completely:
a. 9x4 − 16x2
b. 2x3 + 5x2 − 8x − 20
c. x3 + 3x2 + 3x + 1
d. 8x3 − 1

Answer 1

Answer:

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

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

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

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

$= (3x^2 + 4) ( ((\sqrt{3}x )^2 - (2\sqrt{x})^2)$

$= (3x^2 + 4) (\sqrt{3}x + 2\sqrt{x})(\sqrt{3}x - 2\sqrt{x})$

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. $8x^3 - 1$

$=(2x)^3 - 1$

$=(2x-1)((2x)^2 + 2x\times 1 + 1^2)$

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

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