Question

use polynomial identities to factor (64x^9 + 125y^6).
a. ((4x^2 + 5y)(16x^4 + 20x^2y + 25y^2))
b. ((4x^2 - 5y)(16x^4 - 20x^2y + 25y^2))
c. ((4x^3 + 5y^2)(16x^6 - 20x^3y^2 + 25y^4))
d. ((4x^3 - y^2)(16x^6 + 20x^3y^2 + 25y^4))

Explanation:

Step1: Recognize sum of cubes

Recall identity: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Rewrite $64x^9 + 125y^6$ as $(4x^3)^3 + (5y^2)^3$, so $a=4x^3$, $b=5y^2$.

Step2: Compute $a^2$

$a^2 = (4x^3)^2 = 16x^6$

Step3: Compute $ab$

$ab = (4x^3)(5y^2) = 20x^3y^2$

Step4: Compute $b^2$

$b^2 = (5y^2)^2 = 25y^4$

Step5: Substitute into identity

$(a+b)(a^2 - ab + b^2) = (4x^3 + 5y^2)(16x^6 - 20x^3y^2 + 25y^4)$

Answer:

C. $(4x^{3} + 5y^{2})(16x^{6} - 20x^{3}y^{2} + 25y^{4})$