Exponents and Polynomials

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Exponents and Polynomials
Chapter Ten

• 10.1 Adding and Subtracting Polynomials
• 10.2 Multiplication Properties of Exponents
• 10.3 Multiplying Polynomials
• 10.4 Introduction to Factoring Polynomials

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Adding and Subtracting Polynomials
Section 10.1
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For example,
This polynomial is written in descending powers
of x because the powers of x decrease from left
to right.
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Types of Polynomials

• A monomial is a polynomial with exactly one term.
• A binomial is a polynomial with exactly two
  terms.
• A trinomial is a polynomial with exactly three
  terms.

Martin-Gay, Prealgebra, 5ed
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Adding Polynomials

• To add polynomials, use the commutative and
  associative properties and then combine like
  terms.

Remove parentheses.
Combine like terms.
Simplify.
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Subtracting Polynomials
Recalling the definition of subtraction,
a - b a (-b)
to subtract a polynomial, add its opposite.

• To subtract two polynomials, change the signs of
  the terms of the polynomial being subtracted, and
  then add.

Martin-Gay, Prealgebra, 5ed
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Subtracting Polynomials. . .
Add the opposite.
Remove parentheses.
Combine like terms.
Simplify.
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Helpful Hint
Dont forget to change the sign of each term in
the polynomial being subtracted.
Martin-Gay, Prealgebra, 5ed
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Multiplication Properties of Exponents
Section 10.2
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Product Rule for Exponents
If m and n are positive integers and a is a
real number, then
am ? an a m n
For example,

• In other words, to multiply two exponential
  expressions with the same base, keep the base and
  add the exponents. This is called simplifying the
  exponential expression.

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Usually, an exponent of 1 is not written, so when
no exponent appears, we assume that the exponent
is 1. For example,
2 21 and 7 71.
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Martin-Gay, Prealgebra, 5ed
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These examples will remind you of the difference
between adding and multiplying terms.
Addition
Multiplication
7x 2 5x 2
7x 2 ? 5x 2
12x 2
35x 4
4x 5x 3
4x ? 5x 3
20x 4
4x 5x 3
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Power Rule for Exponents
If m and n are positive integers and a is a
real number, then
(am)n am n
For example,

• In other words, to raise an exponential
  expression to a power, keep the base and multiply
  the exponents.

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Martin-Gay, Prealgebra, 5ed
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Take a moment to make sure that you understand
when to apply the product rule and when to apply
the power rule.
Product Rule ?
Add Exponents
Power Rule ?
Multiply Exponents
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Power of a Product Rule
If n is a positive integer and a and b are
real numbers, then
(ab)n a nb n
For example,

• In other words, to raise a product to a power,
  raise each factor to the power.

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Martin-Gay, Prealgebra, 5ed
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Multiplying Polynomials
Section 10.3
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Multiplying Monomials

• To multiply two monomials use the associative and
  commutative properties and regroup. Remember
  that to multiply exponential expressions with a
  common base, add exponents.

Use the commutative and associative properties.
Use the product rule for exponents.
Simplify.
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Multiplying Monomials by Polynomials

• To multiply a monomial and any polynomial, we use
  the distributive property and properties of
  exponents.

a(b c) a b a c
Martin-Gay, Prealgebra, 5ed
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Multiplying Monomials by Polynomials. . .
Apply the distributive property.
3x(5x2 4)
3x 5x2 3x 4 15x3 12x
4z(2z 2 5z 6)
4z 2z 2 4z 5z 4z (- 6) 8z 3 20z 2
24z
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Martin-Gay, Prealgebra, 5ed
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Multiplying Two Polynomials

• To multiply two binomials, we use a version of
  the distributive property.
• (b c)a b a c a

(x 2)(x 3)
x(x 3) 2(x 3)
x x x 3 2 x 2 3
x 2 3x 2x 6
x 2 5x 6
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To Multiply Two Polynomials Multiply each term of
the first polynomial by each term of the second
polynomial and then combine like terms.
Martin-Gay, Prealgebra, 5ed
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Introduction to Factoring Polynomials
Section 10.4
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Finding the Greatest Common Factor
The first step in factoring a polynomial is to
see whether the terms of the polynomial have a
common factor.
If there is a common factor, we can write the
polynomial as a product by factoring out the
common factor.
We will usually factor out the greatest common
factor (GCF).
Martin-Gay, Prealgebra, 5ed
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Finding the Greatest Common Factor. . .
The greatest common factor (GCF) of a list of
terms is the product of the GCF of the numerical
coefficients and the GCF of the variable factors.
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Helpful Hint
Notice below that the GCF of a list of terms
contains the smallest exponent on each common
variable.
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Martin-Gay, Prealgebra, 5ed
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Factoring Out the Greatest Common Factor
To factor a polynomial

• Do the terms have a greatest common factor other
  than 1?

• If so, factor out the greatest common factor from
  each term by writing each term as a product of
  the greatest common factor and the terms
  remaining factors.

• Use the distributive property to write the
  factored form of the polynomial.

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Factoring Out the Greatest Common Factor
Factoring can be checked by multiplying.
Consider, 5x 10

• The GCF of 5x 10 is 5.

• Factor 5 from each term and write each term as a
  product of 5 and the remaining terms,

• Using the distributive property, write

factored form of polynomial
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Helpful Hint
A factored form of 5x 10 is not
Although the terms have been factored (written as
a product), the polynomial 5x 10 has not been
factored. A factored form of 5x 10 is the
product 5(x 2).
factored terms
5(x 2)
factored polynomial
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Examples of Factored Polynomials
Dont forget the 1.
In this example, factor out -2a rather than 2a
Dont forget the -1.
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