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Common Monomial Factoring 675
Lesson 11-
Common Monomial
Factoring
When two or more numbers are multiplied, the result is a single number. Factoring is the reverse process. In factoring, we begin with a single number and express it as the product of two or more numbers. For example, the product of 7 and 4 is 28. So, factoring 28, we get 28 = 7 · 4. In Lesson 11-3, you multiplied monomials by polynomials to obtain polynomials. In this lesson you will learn how to reverse the process.
If factors are not integers, then every number has infinitely many factors. For example, 8 is not only 4 · 2 and 8 · 1, but also 24 · 1 __ 3 and 2.5 · 3.2. For this reason, in this book all factoring is over the set of integers.
Factoring Monomials
Every expression has itself and the number 1 as a factor. These are called the trivial factors. If a monomial is the product of two or more variables or numbers, then it will have factors other than itself and 1.
Example 1
What are the factors of 49x^3? Solution The factors of 49 are 1, 7, and 49. The monomial factors of x^3 are 1, x, x 2 , and x 3. The factors of 49x^3 are the 12 products of a factor of 49 with a factor of x: 1, 7, 49, x, 7x, 49x, x 2 , 7x 2 , 49x 2 , x 3 , 7x 3 , 49x 3
QY
The greatest common factor (GCF) of two or more monomials is the product of the greatest common factor of the coefficients and the greatest common factors of the variables.
Vocabulary
factoring trivial factors greatest common factor factorization prime polynomials complete factorization
BIG IDEA Common monomial factoring is the process of writing a polynomial as a product of two polynomials, one of which is a monomial that factors each term of the polynomial.
QY Which of the factors of 49 x^3 are trivial factors?
Lesson
Consider the parabola y = 3(x – 6) 2 + 4. a. What is its vertex? b. Does the parabola open up or down? c. True or False The parabola is congruent to y = –3x^2. d. What is an equation for its line of symmetry?
Mental Math
676 Polynomials
Example 2
Find the greatest common factor of 6xy^2 and 18y. Solution The GCF of 6 and 18 is 6. The GCF of xy^2 and y is y. Because the factor x does not appear in all terms, it does not appear in the GCF. So the GCF of 6xy 2 and 18y is 6 · y, which is 6y.
Notice that the GCF of the monomials includes the GCF of the coefficients of the monomials. It also includes any common variables raised to the least exponent of that variable found in the terms.
As with integers, the result of factoring a polynomial is called a factorization. Here is a factorization of 6 x^2 + 12 x.
6 x^2 + 12 x = 2 x (3 x + 6)
Again, as with integers, a factorization with two factors means that a rectangle can be formed with the factors as its dimensions. Here is a picture of the factorization.
Step 1 Build or draw two other rectangles with an area of 6x^2 + 12 x. Step 2 Write the factorization that is shown by each rectangle. Step 3 Do any of the rectangles have the greatest common factor of 6x^2 and 12x as a side length? If so, which rectangle?
The Activity points out that there is more than one way to factor 6 x^2 + 12 x. When factoring a polynomial, the goal is that the GCF of all the terms is one factor. In 6 x^2 + 12 x, 6 x is the greatest common factor, so 6 x^2 + 12 x = 6 x ( x + 2).
Monomials such as 6 x, and polynomials such as x + 2 that cannot be factored into polynomials of a lower degree, are called prime polynomials. To factor a polynomial completely means to factor it into prime polynomials. When there are no common numerical factors in the terms of any of the prime polynomials, the result is called a complete factorization. The complete factorization of 6 x + 12 is 6( x + 2).
Chapter 11
Activity
x^2 x^2 x^2^ x^ x^ x
x^2 x^2 x^2^ x^ x^ x
x
x
x x x^1 1
x x x
x x x
1 1 1
678 Polynomials
Questions
COVERING THE IDEAS
- List all the factors of 33 x^4.
In 2 and 3, find the GCF.
- 25 y^5 and 40 y^2 3. 17 a^2 b^2 and 24 ba^2
- Represent the factorization 12 x^2 + 8 x = 4 x (3 x + 2) with rectangles.
- a. Factor 15 c^2 + 5 c by finding the greatest common factor of the terms. b. Illustrate the factorization by drawing a rectangle whose sides are the factors.
- Showing tiles, draw two different rectangles each with area equal to 16 x^2 + 4 x.
- Explain why x^2 + xy is not a prime polynomial.
- In Parts a–c, complete the products. a. 36 x^3 + 18 x^2 = 6(?^ +?^ ) b. 36 x^3 + 18 x^2 = 18(?^ +?^ ) c. 36 x^3 + 18 x^2 = 18 x^2 (?^ +?^ ) d. Which of the products in Parts a–c is a complete factorization of 36 x^3 + 10 x^2? Explain your answer.
- Simplify 24 n _________^6 +^20 n^4 4 n^2
- Find the greatest common factor of 28 x^5 y^2 , –14 x^4 y^3 , and 49 x^3 y^4.
In 11–14, factor the polynomial completely.
- 33 a - 33 b + 33 ab 12. x 2,100^ - x 2,
- 12 v^9 + 16 v^10 14. 46 cd^3 - 69 cd^2 + 18 c^2 d^2
APPLYING THE MATHEMATICS
- The area of a rectangle is 14 r^2 h. One dimension is 2 r. What is the other dimension?
- The top vertex in the fact triangle at the right has the expression 27 abc - 45 a^2 b^2 c^2. What expression belongs in the position of the question mark?
Chapter 11
÷ × 9 ac
27 abc
- 45 a^2 b^2 c^2
?
Common Monomial Factoring 679
- a. Graph y = 2 x^2 - 8 x. b. Graph y = 2 x ( x - 4). c. What do you notice about the graphs of the equations? Explain why this occurs.
In 18 and 19, a circular cylinder with height h and radius r is pictured at the right. Factor the expression giving its surface area.
- π r^2 + 2 π rh, the surface area with an open top
- 2 π r^2 + 2 π rh, the surface area with a closed top
In 20 and 21, simplify the expression.
(^9) ______________ x^2 y + 54 xy - 9 xy^2 9 xy 21.^
- ___________________ 100 n^100 - 80 n^80 + 60 n^60 2 n^2
REVIEW
In 22 and 23, simplify the expression. (Lesson 11-3)
- – 4 x^3 (3 - 5 x^2 + 7 x^4 ) 23. k ( k + 2 k 2 + n ) - 2 n ( k - 2 n ) - k^2
- Which investment plan is worth more at the end of 10 years if the annual yield is 6%? Justify your answer. (Lesson 11-1) Plan A: Deposit $50 each year on January 2, beginning in 2008. Plan B: Deposit $100 every other year on January 2, beginning in
- Multiple Choice Which system of inequalities describes the shaded region in the graph at the right? (Lesson 10-9)
A
y - x < 6 x ≤ 0 y ≤ 6
B
y + 2 x ≥ 6 x ≤ 6 y ≤ 0
C
y + 2 x ≤ 6 x ≤ 0 y ≤ 0
D
y ≤ 6 + 4 x x ≥ 3 y ≥ 1
26. Simplify √ 50 x^3 y^4. (Lesson 8-6)
- There are 4 boys, 7 girls, 6 men, and 5 women on a community youth board. How many different leadership teams consisting of one adult and one child could be formed from these people? (Lesson 8-1)
EXPLORATION
- The number 6 has four factors: 1, 2, 3, and 6. The number 30 has eight factors: 1, 2, 3, 5, 6, 10, 15, and 30. a. Find five numbers that each have an odd number of factors. b. Give an algebraic expression that describes all numbers with an odd number of factors. Explain why you think these numbers have an odd number of factors.
Lesson 11-
r
h
QY ANSWER 1 and 49x^3
y
x
2
**-
-**
–2 2 4 6 8
