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113NA_Algebra1_Polynomials_2/17 2/24/05 1:40 PM Page 3
Table of Contents
To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Session I: Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
NOTICE: Photocopying any part of this book is forbidden by law.
Session II: Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Lesson 1:
Simplify Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
A. Simplify Expressions with Opposites, Reciprocals, and
Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
B. Simplify Expressions with Exponents and Roots . . . . . . . . . . . . . .22
Lesson 2:
Use Number Properties and Laws of Exponents . . . . . . . . . . . . . . .26
A. Recognize Members of and Describe Characteristics of
Different Sets of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
B. Recognize and Use Real Number Properties . . . . . . . . . . . . . . . . .31
C. Recognize and Use Laws of Exponents . . . . . . . . . . . . . . . . . . . . .37
Lesson 3:
Add and Subtract Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
A. Add Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
B. Subtract Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Lesson 4:
Multiply and Factor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .45
A. Multiply Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
B. Factor Greatest Common Monomials from Polynomials . . . . . . . . .48
Session III: Posttest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51
Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
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Lesson
4
A
Multiply and Factor
Polynomials
Multiply Polynomials
Look at how these monomials are multiplied:
5m2n 6m5n3 30m7n4
Notice that factors of each type can be multiplied separately.
5 6 30
m2 m5 m2 5 m7
n n3 n1 3 n4
Multiply the constants.
Multiply exponents with the same base.
Remember that n n1.
To multiply a monomial by a polynomial, apply the distributive property.
2ab(a2 3b) 2ab a2 2ab 3b 2a3b 6ab2
To multiply two polynomials, multiply each term in the first by each term in the second.
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(2x y)(x 6y) (2x)(x 6y)
2x x 2x 6y
2x2 12xy
(y)(x 6y) (y) x (y) 6y xy 6y2
The final step is to combine like terms.
2x2 12xy xy 6y2 2x2 11xy 6y2
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Skills Coach, Algebra I: Streamline to Proficiency—Polynomials
Coached Practice
Multiply (6pq 2q)(p2 2q).
STRATEGY:
Multiply each term in the first expression by each term in the second
expression. When multiplying binomials, this is called the FOIL
method.
Multiply terms in this order: First terms, Outer terms, Inner terms, Last
terms
First
terms
(a + b)
Last
terms
O
I
L
F
(c + d) = ac + ad + bc + bd
Inner
terms
Outer terms
Apply the FOIL method to this problem.
(6pq 2q)(p2 2q) F
O
I
L
2
2
6pq p 6pq 2q 2q p 2q 2q 6p3q
12pq2
2p2q
4q2
Notice that there are no like terms to combine.
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(6pq 2q)(p2 2q) = 6p3q 12pq2 2p2q 4q2
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SOLUTION:
113NA_Algebra1_Polynomials_2/17 2/24/05 1:40 PM Page 47
Lesson 4: Multiply and Factor Polynomials
Independent Practice
Multiply the polynomials.
1.
6x3y5 2x6y
2.
10a(2a2b 3b3) ______________________________________________________________
3.
4(c3d 2cd2) ________________________________________________________________
4.
(x 2)(4x 1)
5.
(2m2 – 3n)(mn 5n2) __________________________________________________________
_________________________________________________________________
NOTICE: Photocopying any part of this book is forbidden by law.
_____________________________________________________________
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Skills Coach, Algebra I: Streamline to Proficiency—Polynomials
B
Factor Greatest Common Monomials from
Polynomials
In the last lesson you learned how to multiply a monomial by a polynomial using the
distributive property. The reverse process is called factoring.
To factor a polynomial, find the greatest common factor of each term. Remember that the
greatest common factor (GCF) is the greatest number or expression that can evenly
divide each number. For a polynomial, this will sometimes be a monomial.
To find the GCF monomial of a polynomial:
1. Look at the common elements — the expressions that appear in every term.
2. Find the GCF of the coefficients.
3. Find the GCF for each variable. (Use the lowest exponent of the variable.)
4. Multiply these numbers and variables to form the GCF monomial.
For the polynomial
4a3b 2a4b2
Common elements
Terms
GCF
coefficients
variable a
variable b
Combined GCF
4 and 2
a3 and a4
b and b2
2
a3
b
2a3b
So, the GCF of the polynomial 4a3b 2a4b2 is 2a3b.
NOTICE: Photocopying any part of this book is forbidden by law.
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Lesson 4: Multiply and Factor Polynomials
Coached Practice
Factor 16x4y2 8x2y.
STRATEGY:
1. Find the GCF of both terms.
2. Divide each term by the GCF to find the remaining factor of each
monomial.
3. Combine the remaining factors to form the remaining polynomial.
4. Apply the distributive property: rewrite the polynomial as the
product of the GCF and the remaining polynomial.
Find the GCF of the terms.
The
The
The
The
GCF
GCF
GCF
GCF
of
of
of
of
the
the
the
the
coefficients,
common x-variable elements,
common y-variable elements,
polynomial is 8x2y.
16 and 8,
x4 and x2,
y2 and y,
is 8.
is x2.
is y.
Divide each monomial term by the GCF.
16x y
8x y
4 2
2x2y
2
8x y
8x y
2
1
2
Combine the remaining factors using the signs in the polynomial.
Remaining polynomial: 2x2y 1
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Then multiply the remaining polynomial by the GCF.
SOLUTION:
16x4y2 8x2y 8x2y(2x2y 1)
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Skills Coach, Algebra I: Streamline to Proficiency—Polynomials
Independent Practice
Factor each polynomial.
1.
5a + 5b + 5c _________________________________________________________________
2.
2x2 – 6x + 10 _________________________________________________________________
3.
7a2b – 20ab + 3ab2 _____________________________________________________________
4.
15m3n – 10m5n3 _______________________________________________________________
5.
It is said that a polynomial like 10x 3y is in factored form already since it cannot be factored
further. Explain why this polynomial cannot be factored.
NOTICE: Photocopying any part of this book is forbidden by law.
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