POLYNOMIALS

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POLYNOMIALS

• Chapter 4

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4-1 Exponents
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EXPONENTIAL FORM number written such that it
has a base and an exponent
43 4 4 4
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BASE tells what factor is being
multiplied EXPONENT Tells how many equal
factors there are
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EXAMPLES

• x x x x x4
• 6 6 6 63
• 3. -2 p q 3 p q p -6p3q2
• (-2) b (-4) b 8b2

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ORDER OF OPERATIONS

• Simplify expression within grouping symbols
• Simplify powers
• Simplify products and quotients in order from
  left to right
• Simplify sums and differences in order from left
  to right

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EXAMPLES

• -34 -(3)(3)(3)(3) - 81
• (-3)4 (-3)(-3)(-3)(-3) 81
• (1 5)2 (6)2 36
• 1 52 1 25 26

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4-2 Adding and Subtracting Polynomials
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DEFINITIONS

• Monomial an expression that is either a
  numeral, a variable, or the product of a numeral
  and one or more variables.
• -6xy, 14, z, 2/3r, ab

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DEFINITIONS

• Polynomial an expression that is the sum of
  monomials
• 14 2x x2 -4x

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DEFINITIONS

• Binomial an expression that is the sum of two
  monomials (has two terms)
• 14 2x, x2 - 4x

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DEFINITIONS

• Trinomial an expression that is the sum of
  three monomials (has three terms)
• 14 2x y, x2 - 4x 2

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DEFINITIONS

• Coefficient the numeral preceding a variable
• 2x coefficient 2

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DEFINITIONS

• Similar terms two monomials that are exactly
  alike except for their coefficients
• 2x, 4x, -6x, 12x, -x

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DEFINITIONS

• Simplest form when no two terms of a polynomial
  are similar
• 4x3 10x2 2x - 1

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DEFINITIONS

• Degree of a variable the number of times that
  the variable occurs as a factor in the monomial
• 4x2 degree of x is 2

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DEFINITIONS

• Degree of a monomial the sum of the degrees of
  its variables.
• 4x2y degree of monomial is 3

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DEFINITIONS

• Degree of a polynomial is the greatest of the
  degrees of its terms after it has been
  simplified.
• -6x3 3x2 x2 6x3 5

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Examples

• (3x2y4xy2 y33)
• (x2y3y3 4)
• (-a5 5ab4b2 2)
• (3a2 2ab 2b2 7)

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4-3 Multiplying Monomials
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RULE OF EXPONENTS Product of Powers

• am an am n
• x3 x5 x8
• (3n2)(4n4) 12n6

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4-4 Powers of Monomials
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RULE OF EXPONENTS Power of a Power

• (am)n amn
• (x3)5 x15

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RULE OF EXPONENTS Power of a Product

• (ab)m ambm
• (3n2)3 33n6

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4-5 Multiplying Polynomials by Monomials
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Examples Use Distributive Property

• x(x 3) x2 3x
• 4x(2x 3) 8x2 12x
• -2x(4x2 3x 5) -8x36x2 10x

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4-6 Multiplying Polynomials
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Use the Distributive Property

• (3x 2)(2x2- 5x- 4)
• (y 2x)(x3 2y3 3xy2 x2y)

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4-7 Transforming Formulas
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Examples

• C 2?r, solve for r
• c/2? r

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Examples

• S v/r, solve for r
• R v/s

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4-8 Rate-Time-Distance Problems
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Example 1

• A helicopter leaves Central Airport and flies
  north at 180 mi/hr. Twenty minutes later a plane
  leaves the airport and follows the helicopter at
  330 mi/h. How long does it take the plane to
  overtake the helicopter.

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Use a Chart
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Solution

• 330t 180(t 1/3)
• 330t 180t 60
• 150t 60
• t 2/5

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Example 2

• Bicyclists Brent and Jane started at noon from
  points 60 km apart and rode toward each other,
  meeting at 130 PM. Brents speed was 4 km/h
  greater than Janes speed. Find their speeds.

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Use a Chart
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Solution

• 1.5(r 4) 1.5 r 60
• 1.5r 6 1.5r 60
• 3r 6 60
• 3r 54
• r 18

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4-9 Area Problems
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Examples

• A rectangle is 5 cm longer than it is wide. If
  its length and width are both increased by 3 cm,
  its area is increased by 60 cm2. Find the
  dimensions of the original rectangle.

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Draw a Picture
x 5
x
x 3
x 8
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Solution

• x(x5) 60 (x3)(x 8)
• X2 5x 60 x2 11x 24
• 60 6x 24
• 36 6x
• 6 x and 6 5 11

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Example 2

• Hector made a rectangular fish pond surrounded by
  a brick walk 2 m wide. He had enough bricks for
  the area of the walk to be 76 m2. Find the
  dimensions of the pond if it is twice as long as
  it is wide.

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Draw a Picture
2 m
2x
x 4
2 m
x
2x 4
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Solution

• (2x 4)(x 4) (2x)(x) 76
• 2x2 8x 4x 16 2x2 76
• 12x 16 76
• -16 -16
• 12x 60
• 12 12
• x 5

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4-10 Problems Without Solutions
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Examples

• A lawn is 8 m longer than it is wide. It is
  surrounded by a flower bed 5 m wide. Find the
  dimensions of the lawn if the area of the flower
  bed is 140 m2

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Draw a Picture
x 8
x
5
5
x 8
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Solution

• (x10)(x18) x(x8) 140
• x2 28x 180 x2 -8x 140
• 20x -40
• x -2
• Cannot have a negative width

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THE END