CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc....
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Transcript of CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc....
CHAPTER
4Polynomials: Operations
Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4.1 Integers as Exponents
4.2 Exponents and Scientific Notation
4.3 Introduction to Polynomials
4.4 Addition and Subtraction of Polynomials
4.5 Multiplication of Polynomials
4.6 Special Products
4.7 Operations with Polynomials in Several Variables
4.8 Division of Polynomials
OBJECTIVES
4.3 Introduction to Polynomials
(continued)
Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Evaluate a polynomial for a given value of the variable.
b Identify the terms of a polynomial.c Identify the like terms of a polynomial.d Identify the coefficients of a polynomial.e Collect the like terms of a polynomial.f Arrange a polynomial in descending order, or collect
the like terms and then arrange in descending order.
OBJECTIVES
4.3 Introduction to Polynomials
Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
g Identify the degree of each term of a polynomial and the degree of the polynomial.
h Identify the missing terms of a polynomial.i Classify a polynomial as a monomial, a binomial, a
trinomial, or none of these.
Examples:
3x2 2 2x 3x6 0
A monomial is an expression of the type axn, where a is a real number constant and n is a nonnegative integer.
4.3 Introduction to Polynomials
Monomial
Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Examples:5w + 8, 3x2 + x + 4, x, 0, 75y6
A polynomial is a monomial or a combination of sums and/or differences of monomials.
4.3 Introduction to Polynomials
Polynomial
Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
When we replace the variable in a polynomial with a number, the polynomial then represents a number called a value of the polynomial.
Finding that number, or value, is called evaluating the polynomial.
4.3 Introduction to Polynomials
a Evaluate a polynomial for a given value of the variable.
Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLEa. 5x + 2 b. 3x2 – 4x + 1
Solutiona. 5x + 2 = 5 · 3 + 2
= 15 + 2 = 17
b. 3x2 – 4x + 1 = 3 · 32 – 4 · 3 + 1 = 3 · 9 – 4 · 3 + 1 = 27 – 12 + 1 = 16
4.3 Introduction to Polynomials
a Evaluate a polynomial for a given value of the variable.
A Evaluate the polynomial when x = 3.
Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLESolution For x = 3, we have x3 + 4x + 7 = (3)3 + 4(3) + 7 = (27) + 4(3) + 7 = 27 + (12) + 7
= 22
4.3 Introduction to Polynomials
a Evaluate a polynomial for a given value of the variable.
B Evaluate x3 + 4x + 7 for x = 3.
Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLEIn a sports league of n teams in which each team plays every other team twice, the total number of games to be played is given by the polynomial n2 n.
A boys’ soccer league has 12 teams. How many games are played if each team plays every other team twice?
4.3 Introduction to Polynomials
a Evaluate a polynomial for a given value of the variable.
C Applications of Polynomials
(continued)
Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLESolution: We evaluate the polynomial for n = 12:
n2 n = 122 12 = 144 12 = 132.The league plays 132 games.
4.3 Introduction to Polynomials
a Evaluate a polynomial for a given value of the variable.
C Applications of Polynomials
Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
SolutionThe terms are written as 7p5, 3p3, and 3.
7p5 3p3 + 3
These are the terms of the polynomial.
4.3 Introduction to Polynomials
b Identify the terms of a polynomial.
D Identify the terms of the polynomial 7p5 3p3 + 3.
Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
When terms have the same variable and the variable is raised to the same power,
we say that they are like terms.
4.3 Introduction to Polynomials
c Identify the like terms of a polynomial.
Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
a. 5x3 + 6x – 3x2 + 2x3 + x2 b. 8 – 7a2 – 9 – a – 2a Solution
a. 5x3 + 6x – 3x2 + 2x3 + x2
Like terms: 2x3 and 5x3 Same variable and exponentLike terms: –3x2 and x2 Same variable and exponent
b. 8 – 7a2 – 9 – a – 2a Like terms: –a and –2a, 8 and –9
4.3 Introduction to Polynomials
c Identify the like terms of a polynomial.
E Identify all the like terms in the polynomials.
Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The part of a term that is a constant factor is the coefficient of that term.
The coefficient of 4y is 4.
4.3 Introduction to Polynomials
d Identify the coefficients of a polynomial.
Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
5x4 – 8x2y + y – 9Solution
The coefficient of 5x4 is 5.The coefficient of –8x2y is –8.The coefficient of y is 1, since y = 1y.The coefficient of –9 is simply –9.
4.3 Introduction to Polynomials
d Identify the coefficients of a polynomial.
F Identify the coefficient of each term in the polynomial.
Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLEa) 4y4 9y4 b) 7x5 + 9 + 3x2 + 6x2 13 6x5
c) 9w5 7w3 + 11w5 + 2w3
Solutiona) 4y4 9y4 = (4 9)y4 = 5y4
b) 7x5 + 9 + 3x2 + 6x2 13 6x5 = 7x5 6x5 + 3x2 + 6x2 + 9 13= x5 + 9x2 4
4.3 Introduction to Polynomials
e Collect the like terms of a polynomial.
G Combine like terms.
(continued)
Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLEa) 4y4 9y4 b) 7x5 + 9 + 3x2 + 6x2 13 6x5
c) 9w5 7w3 + 11w5 + 2w3
Solutionc) 9w5 7w3 + 11w5 + 2w3 = 9w5 + 11w5 7w3 + 2w3
= 20w5 5w3
4.3 Introduction to Polynomials
e Collect the like terms of a polynomial.
G Combine like terms.
Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
We usually arrange polynomials in descending order by exponent, but not always. The opposite order is called ascending order.
7x5 + 5x7 + x2 + 3x3
Solution: 7x5 + 5x7 + x2 + 3x3 = 5x7 + 7x5 + 3x3 + x2
4.3 Introduction to Polynomials
f Arrange a polynomial in descending order, or collect the like terms and then arrange in descending order.
F Arrange the polynomial in descending order.
Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The degree of a term of a polynomial is the number of variable factors in that term.
4.3 Introduction to Polynomials
g Identify the degree of each term of a polynomial and the degree of the polynomial.
Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
a) 9x5 b) 6y c) 9
Solutiona) The degree of 9x5 is 5.b) The degree of 6y is 1.c) The degree of 9 is 0.
4.3 Introduction to Polynomials
g Identify the degree of each term of a polynomial and the degree of the polynomial.
I Determine the degree of each term:
Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The degree of the polynomial is the largest of the degrees of the terms, unless it is a polynomial 0.
4.3 Introduction to Polynomials
g Identify the degree of each term of a polynomial and the degree of the polynomial.
Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE4x2 9x3 + 6x4 + 8x 7.
Solution The largest exponent is 4.The degree of the polynomial is 4.
4.3 Introduction to Polynomials
g Identify the degree of each term of a polynomial and the degree of the polynomial.
J Identify the degree of the polynomial.
Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
If a coefficient is 0, we generally do no write the term. We say that we have a missing term.
4.3 Introduction to Polynomials
h Identify the missing terms of a polynomial.
Slide 24Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
8x6 – 3x3 + 5x2 + 9x – 8 SolutionThere is no x5 or x4 term so those two terms are missing.
8x6 + 0x5 + 0x4 – 3x3 + 5x2 + 9x – 8
4.3 Introduction to Polynomials
h Identify the missing terms of a polynomial.
K Identify the missing terms in the polynomial.
Slide 25Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
Solution: x4 – 3x3 + 5x – 8 Missing terms: x4 – 3x3 + 0x2 + 5x – 8
Leaving space: x4 – 3x3 + 5x – 8
4.3 Introduction to Polynomials
h Identify the missing terms of a polynomial.
L Write the polynomial x4 – 3x3 + 5x – 8 in two ways: with its missing terms and by leaving space for them.
Slide 26Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Monomials Binomials Trinomials None of These
5x2 3x + 4 3x2 + 5x + 9 5x3 6x2 + 2xy 98 4a5 + 7bc 7x7 9z3 + 5 a4 + 2a3 a2 + 7a 28a23b3 10x3 7 6x2 4x ½ 6x6 4x5 + 2x4 x3 + 3x 2
A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Polynomials with four or more terms have no special name.
4.3 Introduction to Polynomials
i Classify a polynomial as a monomial, a binomial, a trinomial, or none of these.
Slide 27Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
