Roadmap
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Notes
Chapter 4 word problems
Homework questions for §4.2, 5.1
§5.2, 5.3 examples
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Assignment
Notes
For Wednesday:
1. Exercises from 5.2, 5.3
2. Wednesday - Quiz # 5: §4.1 - 4.2, 5.1 - 5.3 and
Cumulative Portion
3. Next Exam: October 30 - Chapter 4 & 5, and
Cumulative Portion
Study guide: For Quiz 5, review exercises on pages
290-291: §4.1 1-25 odd, §4.2 29 - 36, and page 5.1 1-13
odd, 19 - 41 odd.
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Chapter 5: Polynomials - Vocabulary
Notes
Polynomials in x are algebraic expressions of the form:
an xn + an−1 x n−1 + · · · + a2 x 2 + a1 x + a0
where an , an−1 , . . . , a2 , a1 , a0 are real number coefficients.
Each exponent is a whole number. The value n is the
degree. The coefficients an and a0 are special. They are
the leading coefficient and the constant term respectively.
Write a polynomial in standard form with descending
powers of the variable:
Example: 2x 4 − 3x 2 + x − 6.
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§5.2 - Adding and Subtracting Polynomials
Notes
To add or subtract polynomials, combine like terms.
Example: Add
5x 3 + 2x 2 - x - 7
4x 2 - 4x + 6
= (5 + 0)x 3 + (2 + 4)x 2 + (−1 − 4)x + (−7 + 6) =
5x 3 + 6x 2 − 5x − 1
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§5.2 - Adding and Subtracting Polynomials
Notes
Example: Subtract
(x 3 − 6x + 5) − (2x 3 − 3x 2 + 7x − 1)
(x 3 −6x + 5) − (2x 3 − 3x 2 + 7x − 1)
= (x 3 − 6x + 5) + (−2x 3 + 3x 2 − 7x + 1)
= (x 3 − 2x 3 ) + (3x 2 ) + (−6x − 7x) + (5 + 1)
= −x 3 + 3x 2 − 13x + 6
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§5.2 - Adding and Subtracting Polynomials
Notes
Examples:
# 29: Find the sum. (2x 2 − 3) + (5x 2 + 6)
# 43: Find the sum. (5x 2 − 3x + 4) + (−3x 2 − 4)
# 53: Find the difference. (3x 2 − 2x + 1) − (2x 2 + x − 1)
# 63: Find the difference. (x 2 − x + 3) − (x − 2)
# 71: Perform the operations:
(4x 5 − 10x 3 + 6x) − (8x 5 − 3x 3 + 11) + (4x 5 + 5x 3 − x 2 )
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§5.3 - Multiplying Polynomials
Notes
Monomials have only one term.
Example: Multiply the polynomial with the monomial
(applying the Distributive property):
5x 2 (3x − 8)
5x 2 (3x − 8) = 5x 2 (3x) + 5x 2 (−8) = 15x 3 − 40x 2
Example:
(−7x 3 + 15x 2 − 1)(−x 4 )
(−7x 3 + 15x 2 − 1)(−x 4 ) =
(−7x 3 )(−x 4 )+(15x 2 )(−x 4 )+(−1)(−x 4 ) = 7x 7 −15x 6 +x 4
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§5.3 - Multiplying Polynomials
Notes
Multiplying two binomials uses the Distributive Property
(mnemonic is ”FOIL” - First, Outside, Inside, Last).
Example:
(2x + 3)(5x − 1)
(2x + 3)(5x − 1) = 2x(5x − 1) + 3(5x − 1)
= (2x)(5x) + (2x)(−1) + (3)(5x) + (3)(−1)
= 10x 2 − 2x + 15x − 3
= 10x 2 + 13x − 3
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§5.3 - Multiplying Polynomials
Notes
The Distributive Property applies when multiplying
polynomials with more terms than two.
Example:
(x + 2)(−x 2 − 4x + 7)
(x + 2)(−x 2 − 4x + 7) = (x)(−x 2 − 4x + 7) + (2)(−x 2 − 4x + 7)
= (−x 3 − 4x 2 + 7x) + (−2x 2 − 8x + 14)
= −x 3 − 6x 2 − x + 14
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§5.3 - Multiplying Polynomials
Notes
Special Products
The Sum and Difference of The Same Two Terms
(u + v )(u − v ) = u 2 − v 2
Square of a Binomial
(u + v )2 = u 2 + 2uv + v 2
(u − v )2 = u 2 − 2uv + v 2
When squaring a binomials the middle term result will
always be plus/minus twice the product of the two terms.
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§5.3 - Multiplying Polynomials
Notes
Example: (6x − 7)(6x + 7) is 36x 2 − 49. Here’s why:
36x 2 − 49 = (6x − 7)(6x + 7)
= (6x)(6x + 7) + (−7)(6x + 7)
= 36x 2 + 42x − 42x − 49
Example: (5 − 2x)2
(5 − 2x)2 = (5 − 2x)(5 − 2x)
= 25 − 10x − 10x + 4x 2
= 4x 2 − 20x + 25
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§5.3 - Multiplying Polynomials
Notes
Examples:
# 23: (5x − 2)(2x − 6)
# 33: (x − 1)(x 2 − 4x + 6)
# 45: (7x 2 − 14x + 9)(2x + 1)
# 69: (x + 5)2
# 83: (k + 5)3
# 89: (6x m − 5)(2x 2m − 3)
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