SAT Test Prep

CHAPTER 8
ESSENTIAL ALGEBRA I SKILLS

Lesson 5: Factoring

Factoring


To factor means to write as a product (that is, a multiplication). All of the terms in a product are called factors (divisors) of the product.


Example:

There are many ways to factor , or .

Therefore, 1, 2, 3, 4, 6, and 12 are the factors of 12.


Know how to factor a number into prime factors, and how to use those factors to find greatest common factors and least common multiples.


Example:

Two bells, A and B, ring simultaneously, then bell A rings every 168 seconds and bell B rings every 360 seconds. What is the minimum number of seconds between simultaneous rings?

This question is asking for the least common multiple of 168 and 360. The prime factorization of 168 is  and the prime factorization of 360 is . A common multiple must have all of the factors that each of these numbers has, and the smallest of these is . So they ring together every 2,520 seconds.


When factoring polynomials, think of “distribution in reverse.” This means that you can check your factoring by distributing, or FOILing, the factors to make sure that the result is the original expression. For instance, to factor , just think: what common factor must be “distributed” to what other factor to get this expression? Answer:  (Check by distributing.) To factor , just think: what two binomials must be multiplied (by FOILing) to get this expression? Answer:  (check by distributing)


The Law of FOIL:

Example:

Factor 3x2-18x.

Common factor is  (check by distributing)

Factor .

 (check by FOILing)

Factoring Formulas


To factor polynomials, it often helps to know some common factoring formulas:


Example:

Factor .

This is a “difference of squares”: .

Factor .

This is a simple trinomial. Look for two numbers that have a sum of –5 and a product of –14. With a little guessing and checking, you’ll see that –7 and 2 work. So .

The Zero Product Property


Factoring is a great tool for solving equations if it’s used with the zero product property, which says that if the product of a set of numbers is 0, then at least one of the numbers in the set must be 0.


Example:

Solve .

Factor: 

Since their product is 0, either  or , so  or –2.


The only product property is the zero product property.


Example:

 does not imply that . This would mean that , which clearly doesn’t work!

Concept Review 5: Factoring

1. What does it mean to factor a number or expression?

3. What is the zero product property?

Factor and check by FOILing:

FOIL:

Solve by factoring and using the zero product property. (Hint: each equation has two solutions.)

 

 

15. If  and , then 

SAT Practice 5: Factoring

1. Chime A and chime B ring simultaneously at noon. Afterwards, chime A rings every 72 minutes and chime B rings every 54 minutes. What time is it when they next ring simultaneously?

(A) 3:18 pm

(B) 3:24 pm

(C) 3:36 pm

(D) 3:54 pm

(E) 4:16 pm

2. For all real numbers x and y, then 

(A) y2

(B) 0

(C) 7

(D) –14

(E) –28

3. If for all real values of x,, then 

4. In the figure above, if , what is the slope of the line segment?

5. If  and , then 

(A) 4

(B) 6

(C) 8

(D) 9

(E) 16

6. If  and , then what is the value of ?

(A) –20

(B) –12

(C) –8

(D) –5

(E) 0

7. If , then

(D) 3x

8. If  and , then 

(A). 1

9. If , then what is  in terms of x?

(C) x2

Answer Key 5: Factoring

Concept Review 5

1. To write it as a product (result of multiplication).

3. If the product of a set of numbers is 0, then at least one of the numbers must be 0.

5.  and , so the least common multiple is .

 and , so the greatest common factor is .

SAT Practice 5

1. C  and , so the least common multiple is . 216 minutes is 3 hours 36 minutes.

2. E You can solve this one simply by plugging in  and  and evaluating . Or you could do the algebra: 

4. A The slope is “the rise over the run,” which is the difference of the y’s divided by the difference of the x’s:

Or you can just choose values for m and n, like 2 and 1, and evaluate the slope numerically. The slope between (1, 1) and (2, 4) is 3, and the expression in (A) is the only one that gives a value of 3.

7. D Plugging in  gives you , and (D) is the only choice that yields 3. Or:

9. E 

Square both sides:

Add 2: