Exponents and Polynomials

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Exponents and Polynomials

• Chapter 5

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Chapter Sections
5.1 Exponents 5.2 Negative Exponents 5.3
Scientific Notation 5.4 Addition and
Subtraction of Polynomials 5.5 Multiplication
of Polynomials 5.6 Division of Polynomials
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5.1

• Exponents

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Product Rule for Exponents
Example Multiply each expression. a.) x4 x3
x7 b.) k10 k14 k24 c.) d d5 d6
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Quotient Rule for Exponents
Example Divide each expression.
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Zero Exponent Rule

• x0 1, x ? 0

Example Simplify each expression. a.) x0
1 b.) k10 ? k10 k0 1 c.) 45p0 45(1) 45
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Power Rule for Exponents

• (xm)n xm?n

Example Simplify each expression. a.) (x5)3
x15 b.) (xy)4 x4y4 c.) (4x3y2)3 43x9y6
64x9y6
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Expanded Power Rule for Exponents
Example Simplify each expression.
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5.2

• Negative Exponents

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Negative Exponent Rule
Example Simplify each expression.
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Fraction Raised to a Negative Exponent Rule
Example Simplify each expression.
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5.3

• Scientific Notation

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Scientific Notation

• A number written in scientific notation is
  written as a number greater than or equal to 1
  and less than 10 (1? a ? 10) multiplied by some
  power of 10. The exponent on the 10 must be an
  integer.

Example a.) 1.5 x 104 b.) 7.03 x 10-2
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Writing in Scientific Notation

• Move the decimal point to the right of the first
  nonzero digit. This will give a number greater
  than or equal to 1 and less than 10.

• Count the number of places you moved the decimal
  point to obtain the number in step 1. If the
  original number was 10 or greater, the count is
  positive. If the original number was less then
  1, the count is negative.

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Writing in Scientific Notation

• Multiply the number obtained in step 1 by 10
  raised to the count (power) found in step 2.

11,543 1.1543 x 104
Example a.) 0.000723 7.23 x 10-4 b.) 541,000
5.41 x 105
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Converting from Scientific Notation

• Observe the exponent of the power of 10.

• a) If the exponent is positive, move the decimal
  point in the number to the right the same number
  of places as the exponent. (This will result in
  a number greater than or equal to 10.)

b) If the exponent is negative, move the decimal
point in the number to the left the same number
of places as the exponent. (This will result in
a number less than 1.)
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Calculations Using Scientific Notation

• a.) (2 x 10-3)(3 x 102)
• (2 x 3)(10-3 x 102)
• 6 x 10-1 0.6

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5.4

• Addition and Subtraction of Polynomials

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Identifying Polynomials

• A polynomial x is an expression containing the
  sum of a finite number of terms of the form axn,
  for any real number a and any whole number n.

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Identifying Polynomials

• A polynomial is written in descending order (or
  descending powers) of the variable when the
  exponents on the variable decrease from left to
  right.

Example 5x6 4x3 7x 9
A polynomial with one term is called a monomial.
A binomial is a two-termed polynomial. A
trinomial is a three-termed polynomial.
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Identifying Polynomials

• The degree of a term of a polynomial in one
  variable is the exponent on the variable in that
  term.

Example 5x6 (Sixth) 4x3 (Third) 7x (First)
9 (Zero)
The degree of a polynomial is the same as that
of its highest-degree term.
Example 5x6 4x3 7x 9 (Sixth)
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Adding Polynomials

• To add polynomials, combine the like terms of
  the polynomials.

Example a.) (5x 6) (2x 3) 5x 6
2x 3 3x 9 b.) (x2y 6x2 3xy2) (-x2y
12x2 4xy2) x2y 6x2 3xy2 (-x2y)
12x2 4xy2 6x2 xy2
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Subtracting Polynomials

• Use the distributive property to remove
  parentheses. (This will have the effect of
  changing the sign of every term within the
  parentheses of the polynomial being subtracted.)
• (4x3 5x2 8) 4x3 5x2 8
• Combine like terms.
• Example
• (5x 6) (2x 3) 5x 6 2x 3 3x
  3

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5.5

• Multiplication of Polynomials

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Multiplying Polynomials

• To multiply a monomial by a monomial, multiply
  their coefficients and use the product rule of
  exponents.

a.) (5x5)(2x) 5 2 x5 x
10x6 b.) (6x2y2)(3xy4) 6 3 x2 x y2 y4
18x3 y6
To multiply a polynomial by a monomial, use the
distributive property.

a.) 3(x - 4) 3(x) 3(-4) 3x - 12 b.) (-3c2
5c 6)(-6c) (-6c)(-3c2) (-6c)(5c)
(-6c)(-6) 18c3 30c2 36c
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Multiplying Polynomials

• To multiply two binomials, use the distributive
  property so every term in one polynomial is
  multiplied by every term in the other polynomial.

Example a.) (7x 3)(2x 4) (7x 3)(2x)
(7x 3)(4) 14x2 6x 28x 12
14x2 34x 12 b.) (z 2y)(4z 3) (z
2y)(4z) (z 2y)(-3) 4z2 8yz
(-3z) (-6y) 4z2 8yz 3z 6y
A common method used to multiply two binomials
is the FOIL method.
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The FOIL Method
Consider (a b)(c d)
The product of the two binomials is the sum of
these four products (a b)(c d) ac ad
bc bd
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The FOIL Method

• Using the FOIL method, multiply (7x 3)(2x 4) .

14x 2 28x 6x 12 14x 2
34x 12
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Formulas for Special Products

• Product of the Sum and Difference of Two Terms
• (a b)(a b) a2 b2

This special product is also called the
difference of two squares formula.
Example a.) (7x 3) (7x 3) 49x2 - 9 b.) (z3
2y4) (z3 2y4) z6 4y8
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Formulas for Special Products

• Square of Binomials
• (a b)2 (a b)(a b) a2 2ab b2
• (a b)2 (a b)(a b) a2 2ab b2

To square a binomial, add the square of the
first term, twice the product of the terms and
the square of the second term.
Example a.) (5x 3)2 25x2 30x 9 b.)
(z3 12y)2 z6 24yz3 144y2
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5.6

• Division of Polynomials

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Dividing Polynomials

• To divide a polynomial by a polynomial, use the
  same method as when performing long division.

Continued.
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Dividing Polynomials
Example continued