Factoring

To factor an algebraic expression is to rewrite it as a product of two or more expressions, called factors. In general, any expression on the GRE that can be factored should be factored, and any expression that can be unfactored (multiplied out) should be unfactored.

Distributive Rule

The most basic type of factoring involves the distributive rule (also know as factoring out a common factor):

ax + ay = a(x + y)

When this rule is applied from left to right, it is called factoring. When the rule is applied from right to left, it is called distributing.

For example, 3h + 3k = 3(h + k), and 5xy + 45x = 5xy+ 9â€¢5x = 5x(y + 9). The distributive rule can be generalized to any number of terms. For three terms, it looks like ax + ay + az = a(x + y + z).

For example, 2x + 4y + 8 = 2x + 2â€¢ 2y + 2 â€¢ 4 = 2(x + 2y + 4). For another example, x2 y2 + xy3 + y5 = y2(x2 + xy+ y3).

Example 1:
If x Â–â€“ y = 9, then $\left(x-\frac{y}{3}\right)$ - $\left(y-\frac{x}{3}\right)$

(A) Â–â€“4
(B) Â–â€“3
(C) 0
(D) 12
(E) 27

$\left(x-\frac{y}{3}\right)$ - $\left(y-\frac{x}{3}\right)$

By distributing the negative sign =

x - $\frac{y}{3}$ - y + $\frac{x}{3}$

by combining the fractions =
$\frac{4}{3}$x - $\frac{4}{3}$y

by factoring out the common factor $\frac{4}{3}$ =

$\frac{4}{3}$(x - y)

since x Â–â€“ y = 9=
$\frac{4}{3}$(9) = 12

Example 2:

Column A
$\frac{{2}^{20}-{2}^{19}}{{2}^{11}}$

Column B
28

$\frac{{2}^{20}-{2}^{19}}{{2}^{11}}$ = $\frac{{2}^{19+1}-{2}^{19}}{{2}^{11}}$ =

by the rule xa â€¢ xb = xa +b =
$\frac{{2}^{19}Â·{2}^{1}-{2}^{19}}{{2}^{11}}$

by the distributive property ax + ay = a(x + y) =
$\frac{{2}^{19}\left(2-1\right)}{{2}^{11}}$ =

$\frac{{2}^{19}}{{2}^{11}}$

by the rule $\frac{{x}^{a}}{{x}^{b}}$ = xa-b =
28
Hence, the columns are equal, and the answer is (C).