A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
A.3 Polynomials & Rational Expressions
Day 1
Objectives:
• Recognize Special Products
• Factor Polynomials
• Simplify Rational Expressions
• Multiply Rational Expressions
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
What is a polynomial?
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
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
monomial
2. 2x + 3
binomial
3. 4y3 + 2y2 + 6
trinomial
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Name the parts of this expression.
5x3
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
More Definitions:
Coefficient: The constant in front of the variable
Example: Given 3x2 the coefficient is 3.
Degree: The highest power of a polynomial.
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Skills Review:
Example 1:
Conjugates
Example 2:
Squaring a Binomial
(3x - 2)(3x + 2)
(2y - 5)2
9x2 + 6x - 6x - 4
4y2 - 20y + 25
9x2 - 4
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Skills Review:
Example 3:
Foil or Distribute
(x - 4)(3x + 2)
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3x + 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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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
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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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
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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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
What is a rational expression?
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
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
x
2. x2-3x + 2
x3-1
3. x
x2-5
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Example 8: Simplifying Rational Expressions
Simplify
x2 + 10x + 25
. State any restrictions on the variable.
2
x + 9x + 20
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Example 9: Simplify each expression. State any restrictions on the variables.
3
2
­27x
y
c. 2x ­ 3x ­ 2
b. ­6 ­ 3x
a.
9x4y
x2 ­ 6x + 8
x2 ­ 5x + 6
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Example 10: Multiply
x2 - 2x + 1
x3 + x
.
4x2 + 4
x2 + x - 2
(x - 1)(x - 1)
x(x2 + 1)
.
4(x2 + 1)
(x + 2)(x - 1)
(x - 1)
x
.
4
(x + 2)
4(x - 1)
x(x + 2)
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Example 11: Multiplying Rational Expressions
2
2
­ 16
x
+ 7x + 3
2x
and . Multiply
x2 + 8x + 15
x ­ 4
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A.3 Day 1 Polynomials and Rational Expressions 2011 August 24, 2011
Homework:
page 975 (16, 17, 21, 27, 29,
33, 40, 45, 49, 59, 63, 67 - 70)
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