5.4 Polynomials

1
5.4 Polynomials

• Algebra 2
• Mrs. Spitz
• Fall 2006

2
Objectives

• Add polynomials,
• Subtract polynomials, and
• Multiply polynomials

3
Assignment

• pp. 225-226 4-67 every 3rd problem
• 4, 7,10,13,16,19,22,25,28,31,34,37,40, 43,46,
  49, 52, 55, 58, 61, 64, 67
• Check your odd answers in the back of the book

4
Introduction

• The expression y2 4x2 is a polynomial. A
  polynomial is a monomial or a sum of monomials.
  What isnt a polynomial?

6x-2 NOT a polynomial term It has a negative exponent
NOT a polynomial term It has a variable in the denominator.
NOT a polynomial term This has a variable inside the radical
4x2 Polynomial Term
5
Degree

• The degree of a polynomial is the sum of the
  exponents of its variables. The degree of a
  polynomial is the greatest of the degrees of its
  terms.

Monomial/Polynomial Degree
5x2 2
4ab3c4 1 3 4 8
-9 0 no variables
8x3 2x2 7 3, 2, and 0, so 3
6x2y 5x3y2z x x2y2 3, 6, 1, and 4, so 6
6
Ex. 1 Find the degree of 8a3 5a2b2 4ab2 b
- 1

• 8a3 has degree 3
• (-5a2b2) has degree 4
• ( 4ab2)has degree 3
• b has degree 1
• - 1 has degree 0
• The highest degree is 4. The degree of the
  polynomial is 4.

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Ex. 2 Simplify 4x2y 2xy3 y 6xy3 x2y 9y

• 4x2y 2xy3 y 6xy3 x2y 9y
• First rearrange so the like terms are next to
  each other.
• (4x2y x2y) ( 2xy3 6xy3) (y 9y)
• (4 1)x2y (-2 6)xy3 (1 9)y
• 3x2y 4xy3 10y

8
Note

• We can simplify a polynomial using the
  distributive property, as we have just seen, or
  by adding or subtracting the coefficients of like
  terms. Study Ex. 3 to see how to simplify by
  adding or subtracting coefficients.

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Ex. 3 Simplify (4a2 7st 2t2) (2s2 8st
5t2)

• (4a2 7st 2t2) (2s2 8st 5t2)
• Distribute the negative to the second part
• (4a2 7st 2t2) (-2s2 - 8st 5t2)
• Next group like terms
• 4a2 - 2s2 7st - 8st 2t2 5t2
• 2s2 st 3t2

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Ex. 4 Find 5a(5ab2 2a2b2 9b)

• The distributive property is also useful in
  multiplying monomials.
• 5a(5ab2 2a2b2 9b)
• 5a ? 5ab2 5a ? 2a2b2 -5a ? 9b
• 25a2b2 10a3b2 45ab

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Ex. 5 Find (x 2)(x 10)

• You may remember how to foil a binomial (FOIL)
• F first terms
• O outer terms
• I inner terms
• L last terms
• (x 2)(x 10)
• (x ? x) (x ? 10) (x
  ? 2) (2 ? 10)
• x2 10x 2x 20
• x2 12x 20

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FOIL Method

• The FOIL Method is an application of the
  distributive property that makes multiplying
  binomials faster.

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Ex. 6 Use the FOIL Method to find (9a 3)(a4)

• (9a 3)(a4)
• 9a ? a 9a ? 4 -3 ? a -3 ? 4
• 9a2 36a 3a 12
• 9a2 33a 12

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Ex. 7 Show geometrically that (a b)2 a2
2ab b2
The area of the square can be expressed by (a
b)2 or by a2 2ab b2
b
b2
ab
ab
a
a2
a
b
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Ex. 8 Find (2x2 10x 4)(x 12)

• (2x2 10x 4)(x 12)
• (2x2 10x 4)x (2x2 10x 4)12
  Distributive
• 2x2 ? x 10x ? x 4 ? x 2x2 ? 12 10x ? 12
  4 ? 12
• 2x3 10x2 4x 24x2 120x 48
• 2x3 10x2 24x2 4x 120x 48
• 2x3 14x2 124x 48