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A.3 Day 1 Polynomials and Rational Expressions 2011
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August 24, 2011
A.3 Polynomials & Rational ExpressionsDay 1
Objectives:• Recognize Special Products• Factor Polynomials• Simplify Rational Expressions• Multiply Rational Expressions
A.3 Day 1 Polynomials and Rational Expressions 2011
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What is a polynomial?
A.3 Day 1 Polynomials and Rational Expressions 2011
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Definitions:
Polynomials: The sum of 2 or more monomials*Polynomials are usually formed by adding, subtracting, multiplying or dividing.
Monomials: The product of a constant and a variable with a nonnegative integer power
Examples:1. 3x2 2. 2x + 3 3. 4y3 + 2y2 + 6
monomial binomial trinomial
A.3 Day 1 Polynomials and Rational Expressions 2011
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Name the parts of this expression.
5x3
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More Definitions:
Coefficient: The constant in front of the variable
Example: Given 3x2 the coefficient is 3.
Degree: The highest power of a polynomial.
A.3 Day 1 Polynomials and Rational Expressions 2011
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Skills Review:
Example 1: Conjugates
(3x - 2)(3x + 2)
9x2 + 6x - 6x - 4
9x2 - 4
Example 2: Squaring a Binomial
(2y - 5)2
4y2 - 20y + 25
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Skills Review:
Example 3: Foil or Distribute
(x - 4)(3x + 2)
3x2 + 2x - 12x - 8
3x2 - 10x - 8
Example 4: Cubing a Binomial
(x - 3)3
(x - 3)(x - 3)(x - 3)
(x2 - 6x + 9)(x - 3)
x3 - 3x2 - 6x2+ 18x + 9x - 27
x3 - 9x2 + 27x - 27
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Skills Review:*Many polynomials can be simplified to factored form.
Example 5: Difference of Two Squares
16x2 - 49
(4x + 7)(4x - 7)
Example 6:
Perfect Square Trinomials
25x2 - 30x + 9
(5x - 3)2
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Skills Review:
Example 7: Factor by Grouping
x3 - 3x2 - x + 3
(x3 - 3x2) + (-x + 3)
x2(x - 3) - 1(x - 3)
(x - 3)(x2 - 1)
(x - 3)(x + 1)(x - 1)
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What is a rational expression?
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Definition:
Rational Expressions: The quoteint of two polynomials.These expressions should be expressed in lowest terms. (numerator and denominator have no common factors)
Examples:1. x2-2 2. x2-3x + 2 3. x
x x3-1 x2-5
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Example 8: Simplifying Rational Expressions
Simplify . State any restrictions on the variable.x2 + 10x + 25 x2 + 9x + 20
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Example 9: Simplify each expression. State any restrictions on the variables.
a. b. c.27x3y 9x4y
6 3xx2 6x + 8
2x2 3x 2x2 5x + 6
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x2 - 2x + 1 4x2 + 4 x3 + x x2 + x - 2
(x - 1)(x - 1) 4(x2 + 1) x(x2 + 1) (x + 2)(x - 1)
(x - 1) 4 x (x + 2)
4(x - 1)x(x + 2)
.
.
.
Example 10: Multiply
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Example 11: Multiplying Rational Expressions
Multiply and .
2x2 + 7x + 3 x 4
x2 16x2 + 8x + 15
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Homework:page 975 (16, 17, 21, 27, 29, 33, 40, 45, 49, 59, 63, 67 - 70)
