Cracking the SAT subject test in math 2, 2nd edition

Cracking the SAT Subject Test in Math 2, 2nd Edition 5 If , then x = 6 If |2m + 5|=23, then m= 7 If , then r = 8 If , then x = 9 If and y ≠ 0, then y= FACTORING AND DISTRIBUTING When manipulating alge[.]

5 If , then x =

6 If |2m + 5|=23, then m=

7 If , then r =

8 If , then x =

9 If and y ≠ 0, then y=

FACTORING AND DISTRIBUTING

When manipulating algebraic equations, you’ll need to use the tools of factoring and distributing These are simply ways of rearranging equations to make them easier to work with

Factoring

Factoring simply means finding some factor that is in every term of an expression and “pulling it out.” By “pulling it out,” we mean dividing each individual term by that factor, and then placing the whole expression in parentheses with that factor on the outside Here’s an example:

x3 − 5x2 + 6x = 0

On the left side of this equation, every term contains at least one x—that

is, x is a factor of every term in the expression That means you can factor out an x:

x3 − 5x2 + 6x = 0

x(x2 − 5x + 6) = 0

The new expression has exactly the same value as the old one; it’s just written differently, in a way that might make your calculations easier Numbers as well as variables can be factored out, as seen in the example below

11x2 + 88x + 176 = 0

This equation is, at first glance, a bit of a headache It’d be nice to get rid

of that coefficient in front of the x2 term In a case like this, check the other terms and see if they share a factor In fact, in this equation, every term on the left side is a multiple of 11 Because 11 is a factor of each term, you can pull it out:

11x2 + 88x +176 = 0 11(x2 + 8x +16) = 0

x2 + 8x +16 = 0 (x + 4)2 = 0

x = −4

As you can see, factoring can make an equation easier to solve

Distributing

Distributing is factoring in reverse When an entire expression in parentheses is being multiplied by some factor, you can “distribute” the factor into each term, and get rid of the parentheses For example:

3x(4 + 2x) = 6x2 + 36

On the left side of this equation the parentheses make it difficult to combine terms and simplify the equation You can get rid of the parentheses by distributing

DRILL 2: FACTORING AND DISTRIBUTING

Practice a little factoring and distributing in the following examples, and keep an eye out for equations that could be simplified by this kind of rearrangement The answers can be found in Part IV

3 If (11x)(50) + (50x)(29) = 4,000, then x =

(A) 2,000 (B) 200 (C) 20 (D) 2 (E) 0.2

5 If ab ≠ 0;

(A) −3 (B) −2 (C) 0 (D) 1 (E) 3

22 If x ≠ −1,