Exponents and Polynomials

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Exponents and Polynomials

• 5-1 Integer Exponents and Scientific Notation
• 5-2 Adding and Subtracting Polynomials
• 5-4 Multiplying Polynomials
• 5-5 Dividing Polynomials

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5-1 Integer Exponents and Scientific Notation

• Using the product rule for exponents.
• If m and n are natural numbers and a is any real
  number, then
• am an amn
• Note a is referred to as the base and m or n is
  the exponent
• Example 1 Use the product rule for exponents, if
  possible.
• A) m8 m6 m14
• B) m5 p4 (rule does not apply since bases
  are different)
• C) (-5p4) (-9p5) 45p9
• D) (-3x2y3) (7xy4) (-21x3y7)
• Note Be careful not to multiply the bases 35
  34 ? 913
• (33333)(3333) 39
• Also 45 34 ? (4 3)54

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5-1 Integer Exponents and Scientific Notation

• Defining zero exponents.
• If a is any non-zero real number, then a0 1
  
• Note 00 is undefined
• Example 2 Evaluate each expression
• A) 290 1
• B) (-29)0 1
• C) -290 -1
• D) 80 - 150 0

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5-1 Integer Exponents and Scientific Notation

• Defining negative exponents.
• For any natural number n and any non-zero real
  number a, then
• Note A negative exponent does not indicate a
  negative number. Negative exponents lead to
  reciprocals
• However
• Example 3 Evaluate each expression

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5-1 Integer Exponents and Scientific Notation

• Defining negative exponents.
• Special rules for negative exponents If a ? 0
  and b ? 0, then
• and
• Example 4 Evaluate each expression

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5-1 Integer Exponents and Scientific Notation

• Using the quotient rule for exponents.
• If a is any non-zero real number and m and n are
  integers, then
• Example 5 Evaluate each expression

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5-1 Integer Exponents and Scientific Notation

• Using the power rules for exponents.
• If a and b are real numbers and m and n are
  integers, then
• Example 6 Evaluate each expression

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5-1 Integer Exponents and Scientific Notation

• Summary of rules for exponents.
• Product Rule
• Quotient Rule
• Zero Exponent
• Negative Exponent
• Power Rules

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5-1 Integer Exponents and Scientific Notation

• Simplifying exponential expressions.
• Example 7 Evaluate each expression

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5-1 Integer Exponents and Scientific Notation

• Using the rules for exponents with Scientific
  Notation.
• A number is written in scientific notation when
  it is expressed in the form
• Example 8 Write each number in scientific form
• Example 9 Write each scientific number in
  standard form

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5-1 Integer Exponents and Scientific Notation

• Using the rules for exponents with Scientific
  Notation.
• Example 9 Evaluate
• Example 10 The distance to the moon is 2.36 x
  105 miles. How long does it take a rocket
  traveling at 3.2 x 105 mph to reach the moon?

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5-2 Adding and Subtracting Polynomials

• Knowing the basic definitions for polynomials.
• A polynomial is a term or a finite sum of terms
  in which all variables have whole number
  exponents and no variables appear in
  denominators.
• Example 1 Examples of polynomials
• A) 3x -5
• B) 4m3 - 5m3p 8
• C) -5t2s3
• Note A polynomial containing only the variable
  x is called a polynomial in x.
• Example 2 Write the polynomial in descending
  powers of the variable.
• -3x4 6x 2x3 x5 x5 - 3x4 2x3 - 6x

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5-2 Adding and Subtracting Polynomials

• Knowing the basic definitions for polynomials.
• Some polynomials are so common they are given
  special names. A polynomial with exactly one term
  is called a monomial. A polynomial with exactly
  two terms is called a binomial. A polynomial with
  exactly three terms is called a trinomial.
• Example 3 Examples of special polynomials
• A) monomials 5x, 5m3, -8, 5m3p
• B) binomials -3x4 6x, 11y 8, 4m3 - 5m3p
• C) trinomials -3x4 6x 2x, 4m3 - 5m3p 8,
  x5 - 3x4 2x3

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5-2 Adding and Subtracting Polynomials

• Finding the degree of a polynomial.
• The degree of a term with one variable is the
  exponent on the variable. The degree of a term
  with more than one variable is defined to be the
  sum of the exponent on the variables. The
  greatest degree of any term in a polynomial is
  called the degree of the polynomial.
• Example 4 The degree of -7t2s3 is (2 3) or 5
• Note The greatest degree of any term in a
  polynomial is called the degree of the
  polynomial
• Example 5 The greatest degree of -3x4 6x 2x3
  x5 is
• 5 since x5 has the highest degree of all the
  terms

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5-2 Adding and Subtracting Polynomials

• Finding the degree of a polynomial.
• Example 6 Find the degree of
• A) -3x4 6x 2x3 degree 4
• B) 3x4 6x 2x3 x5 degree 5
• C) 6x degree 1
• D) -2 degree 0
• E) 4m3 - 32m3p6 degree 9

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5-2 Adding and Subtracting Polynomials

• Adding and Subtracting polynomials.
• Combining like terms means finding terms
  containing exactly the same variables and the
  same powers.
• Example 7 Combine like terms of 2z4 3x4 z4
  - 9x4
• Answer 3z4 - 6x4

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5-2 Adding and Subtracting Polynomials

• Adding and Subtracting polynomials.
• To add two polynomials, combine like terms.
• Note The polynomials may have to be put in
  descending order to recognize the same terms.
• Example 8 Add
• (-5p3 6p2) (-12p2 8p3) 3p3 - 6p2
• Example 9 Add

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5-2 Adding and Subtracting Polynomials

• Adding and Subtracting polynomials.
• To subtract two polynomials, add the first
  polynomial to the negative of the second
  polynomial.
• Note The polynomials may have to be put in
  descending order to recognize the same terms.
• Example 10 Subtract
• (-5p3 6p2) - (-12p2 8p3)
• (-5p3 6p2) (12p2 - 8p3) -13p3 18p2
• Example 11 Subtract

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5-4 Multiplying Polynomials

• Multiplying terms.
• Use the Commutative and Associative properties
  to find the product of two terms
• Example 1 Multiply 8k3y(9ky3) 72k4y4
• Example 2 Multiply 2m2z4(8m3z2) 16m5z6

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5-4 Multiplying Polynomials

• Multiplying any two polynomials.
• Use the Distributive property to extend the
  multiplying of terms to find the product of any
  two polynomials.
• Example 3 Multiply -2r(9r - 5) -18r2 10r
• Example 4 Multiply (2k - 5m) (3k 2m)
• Distributive Property (2k - 5m) (3k)
  (2k - 5m) (2m)
• 6k2 - 15mk 4km - 10m2
• 6k2 - 15mk 4mk - 10m2
• 6k2 - 11mk - 10m2

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5-4 Multiplying Polynomials

• Multiplying any two polynomials.
• Use the Distributive property to extend the
  multiplying of terms to find the product of any
  two polynomials.
• Example 5 Multiply 5a3 - 6a2 2a - 3
• 2a - 5
• (-5)(5a3 - 6a2 2a - 3) - 25a3 30a2 -
  10a 15
• (2a)(5a3 - 6a2 2a - 3) 10a4 - 12a3
  4a2 - 6a
• 10a4 - 37a3 34a2 -16a 15

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5-4 Multiplying Polynomials

• Multiplying any two binomials.
• Use the First, Outer, Inner, Last (FOIL) Method
• First Last
• Example 6 Multiply (5x -3)(2x -5)
• Inner
• Outer
• First 10x2
• Outer -25x
• Inner -6x
• Last 15
• Add 10x2 - 25x - 6x 15 10x2 - 31x 15

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5-4 Multiplying Polynomials

• Multiplying any two binomials.
• Use the First, Outer, Inner, Last (FOIL) Method
• Example 7 Multiply (4y - z)(2y 3z)
• Add 8y2 12yz - 2zy -3z2 8y2 10yz - 3z2
  
• Note - 2zy -2yz
• Example 8 Multiply (4x - 5)(x 3)
• Add 4x2 12x - 5x - 15 4x2 7x - 15

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5-4 Multiplying Polynomials

• Finding the product of the sum and difference of
  two terms.
• The product of the sum and difference of two
  terms x and y is the difference of the sum of the
  squares of the terms
• (x y)(x - y) x2 - y2
• Example 9 Multiply (y 5)(y - 5)
• y2 - 25
• Example 10 Multiply (x - 4y)(x 4y)
• x2 - 16y2

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5-4 Multiplying Polynomials

• Finding the square of a binomial.
• The square of a binomial is the sum of the
  square of the first term, twice the product of
  the two terms, and the square of the last term
• (x y)2 x2 2xy y2
• (x - y)2 x2 - 2xy y2
• Example11 Multiply (y 5)2
• y2 10y 25
• Example 12 Multiply (x - 4y)2
• x2 - 8xy 16y2
• Note (x y)2 ? x2 y2

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5-4 Multiplying Polynomials

• Finding the product of more complicated
  binomials.
• Example 13 Multiply (x - y z)(x - y - z)
• Product of sum and difference ((x-y)
  z)((x-y) - z)
• (x-y)2 - z2
• x2 - 2xy y2 - z2
• Example 14 Multiply (x y)3
• (x y)2 (x y)
• (x2 2xy y2)(x y)
• x3 2x2y xy2 x2y 2xy2 y3
• x3 3x2y 3xy2 y3

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5-4 Multiplying Polynomials

• Multiplying polynomial functions.
• If f(x) and g(x) define functions,
• then (fg)(x) f(x)g(x)
• Note The domain of the product function is the
  intersection of the domains of f(x) and g(x)
• Example 15 For f(x) 3x 1 and g(x) 2x - 5
• Find (fg)(x) and (fg)(2)
• (fg)(x) f(x)g(x)
• (fg)(x) (3x 1)(2x -5)
• (fg)(x) 6x2 -13x -5
• (fg)(2) 6(2)2 -13(2) -5
• (fg)(2) 24 - 26 - 5 -7

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5-5 Dividing Polynomials

• Dividing a polynomial by a monomial.
• To divide a polynomial by a monomial, divide
  each term in the polynomial by the monomial and
  then write each quotient in lowest terms.
• Example 1 6x2 12x 18
• 2x
• Solution
• Example 2 8a2b2 - 20ab3
• 4a3b
• Solution

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5-5 Dividing Polynomials

• Dividing a polynomial by a polynomial of two or
  more terms.
• To divide a polynomial by another polynomial,
  follow the process for dividing two numbers.
• Example 3

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5-5 Dividing Polynomials

• Dividing a polynomial with a missing term.
• Be sure both polynomials are in descending
  order Add a term with a 0 coefficient as a
  placeholder. Example 4

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5-5 Dividing Polynomials

• Dividing polynomial functions. If f(x) and g(x)
  define functions, then (f/g)(x) f(x)/g(x) Note
  The domain of the product function is the
  intersection of the domains of f(x) and g(x)
  excluding values of x where g(x) goes to
  zero. Example 5 For f(x) 2x2 17x 30 and g(x)
  2x 5
• Find (f/g)(x) and (f/g)(-1)