Factor the polynomial $7 x ^ { 3 } - 14 x ^ { 2 } y + 2 x y ^ { 2 } - 4 y ^ { 3 }$ by grouping.
A) $( 7 x - y ) \left( x - 4 y ^ { 3 } \right)$
B) $( 7 x - y ) \left( x + 4 y ^ { 3 } \right)$
C) $( x - 2 y ) \left( 7 x ^ { 2 } - 2 y ^ { 2 } \right)$
D) $( x - 2 y ) \left( 7 x ^ { 2 } + 2 y ^ { 2 } \right)$
E) $( x - 14 y ) \left( 7 x ^ { 2 } - 4 y ^ { 2 } \right)$

Question

Quizplus · Accepted Answer

The first step in grouping is to factor out the greatest common factor from the entire polynomial:

$$7 x ^ { 3 } - 14 x ^ { 2 } y + 2 x y ^ { 2 } - 4 y ^ { 3 } = 7x(x^2-2xy^2)+2y^2(-2x+2y)$$

Now we can group the two terms on the left that have a common factor of $x$ and factor out that common factor, and group the two terms on the right that have a common factor of $2y^2$ and factor out that common factor:

$$7x(x^2-2xy^2)+2y^2(-2x+2y) = x(7x^2-14xy^2)+2y^2(-2x+2y)$$

Now we can factor out an additional factor of $x-2y$ from each of the terms in the expression:

$$x(7x^2-14xy^2)+2y^2(-2x+2y) = (x-2y)(7x^2+2y^2)$$

So the polynomial factors as $(x-2y)(7x^2+2y^2)$, which corresponds to choice D.

Factor the Polynomial 7x3−14x2y+2xy2−4y37 x ^ { 3 } - 14 x ^ { 2 } y + 2 x y ^ { 2 } - 4 y ^ { 3 }7x3−14x2y+2xy2−4y3

Factor the Polynomial $7 x ^ { 3 } - 14 x ^ { 2 } y + 2 x y ^ { 2 } - 4 y ^ { 3 }$