copyright © 2015, 2008, 2011 pearson education, inc. section 6.1, slide 1 chapter 6 polynomial...

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.1, Slide 1

Chapter 6Polynomial Functions


6.1 Adding and Subtracting Polynomial Expressions

and Functions


Polynomials

• A term is a constant, a variable, or a product of a constant and one or more variables raised to powers. Examples:

• A monomial is a constant, a variable, or a product of a constant and one or more variables raised to counting-number powers. Examples:

Definitions

6 1 2 3 53 , , 3, , and 2x x x x y

6 7 43 , , 3, and 5x x x y


Polynomials

Definitions• A polynomial, or polynomial expression, is a

monomial or a sum of monomials.Examples:5x3 – 2x2 + 7x – 4, 4x5y2 – x2, 4x + 1, 5, x, –2x3

• We write polynomials in descending order.• The degree of a term in one variable is the

exponent on the variable. 7x4 is degree 4.


Polynomials

Definition• The degree of a polynomial is the largest degree of

any nonzero term of the polynomial.4x5 – 9x3 + 1 has degree 5.


Example: Describing Polynomials

Use words such as linear, quadratic, cubic, polynomial, degree, one variable, and two variables to describe the expression.

1. –3x2 + 8x – 42. 5x3 – 2x2 + 9x + 13. 6a5b2 – 9a3b3 – 2ab4


Solution1. –3x2 + 8x – 4

The term –3x2 has degree 2, which is larger than the degrees of the other terms. So, the expression is a quadratic (second-degree) polynomial in one variable.

2. 5x3 – 2x2 + 9x + 1The terms 5x3 has degree 3, which is larger than the degrees of the other terms. So, the expression is a cubic (third-degree) polynomial in one variable.


Solution

3. 6a5b2 – 9a3b3 – 2ab4

The term 6a5b2 has degree 7 (the sum of the exponents of the variables), which is larger than the degrees of the other terms. So, the expression is a seventh-degree polynomial in two variables.


Combining Like Terms

Definitions• The coefficient of a term is the constant factor of

the term.Example: For the term –7x3, the coefficient is –7.

• The leading coefficient of a polynomial is the coefficient of the term with the largest degree.Example: For 3x3 + 6x2 – 4x – 9, the leading coefficient is 3.

• The terms 6x5 and 8x2 are unlike terms (not like terms), because the exponents of x are different.


Combining Like Terms

To combine like terms, add the coefficients of the terms.


Example: Combining Like Terms

Combine like terms when possible.

1. 5x3 – 4x2 + 2x3 – x2

2. 3p3t2 + p2t – 8p3t2 + 7p2t


Solution

1. Rearrange the terms so the terms with x3 are adjacent and the terms with x2 are adjacent:

2.

2 2 2 23 3 3 35 2 5 24 4x xx x xx x x 237 5xx

2 23 2 3 23 8 7p t p ttt pp 2 23 2 3 23 8 7p t p t p t p t

3 2 285 pp tt


Adding Polynomials

To add polynomials, combine like terms.


Example: Adding Polynomials

Find the sum 2 2 2 25 7 3 2 4 9a ab b a ab b


Solution

2 22 27 45 23 9bb baa ba a 2 22 2 72 4 3 95 b bab aba a

2 211 67 ab ba


Example: Subtracting Polynomials

Find the difference 2 27 2 5 6 4 1x x x x


Solution

2 27 2 5 6 4 1x x x x

2 27 2 115 6 4x x x x

2 257 6 12 4x xx x 2 2 2 4 16 57 x xx x

2 42xx


Solution

Use a graphing calculator table to verify the work.


Subtracting Polynomials

To subtract polynomials, first distribute –1; then combine like terms.


Quadratic Function

Definition

A quadratic function is a function whose equation can be put into the form

f(x) = ax2 + bx + c

where a ≠ 0. This form is called the standard form.


Quadratic Function

The graph of a quadratic function is called a parabola. The minimum point or maximum point is called the vertex. The vertical line that passes through a parabola’s vertex is called the axis of symmetry. The part of the parabola that lies to the left of the axis of symmetry is the mirror reflection of the part that lies to the right. Parabolas are illustrated on the next slide.


Parabolas


Example: Using a Graph to Find Values of a Function

A graph of a quadratic function f is sketched below.

1. Find f(4).2. Find x when f(x) = –3.3. Find x when f(x) = 5.4. Find x when f(x) = 6.


Solution

1. The blue arrows show that the input x = 4 leads to the output y = 3. So, f(4) = 3.


Solution

2. The red arrows show that the output y = –3 originates from the two inputs x = –2 and x = 6. So, the values of x are –2 and 6.


Solution

3. The green arrows show that the output y = 5 originates from the single input x = 2. So, the value of x is 2. There is a single input because the vertex (2, 5) is theonly point on the parabola that has a y-coordinate equal to 5.


Solution

4. No point of the downward-opening parabola is above the vertex, which has a y-coordinate of 5. So, there is no point on the parabola with f(x) = 6.


Cubic Function

Definition

A cubic function is a function whose equation can be put into the form

f(x) = ax3 + bx2 + cx + d

where a ≠ 0.


Example: Graphing a Cubic Function

Sketch the graph of f(x) = x3.


Solution

List some input-output pairs of the function, then plot the corresponding points and sketch a curve.


Sum Function, Difference Function

Definition

If f and g are functions and x is in the domain of both functions, then we can form the following functions:

• Sum function f + g, where (f + g)(x) = f(x) + g(x)• Difference function f – g, where

(f – g)(x) = f(x) – g(x)


Modeling with Sum Functions and Difference Functions

Suppose A and B represent quantities. Then A + B represents the sum of the quantities

The difference A – B tells us how much more there is of one quantity than the other quantity.


The Meaning of the Sign of a Difference

If a difference A – B is positive, then A is more than B.

If a difference A – B is negative, then A is less than B.


Example: Using a Sum Function and a Difference Function to Model a Situation

Women’s and men’s enrollments at U.S. colleges and universities are shown in the table on the next slide for various years. The enrollments (in millions) W(t) and M(t) for women and men, respectively, are modeled by the system below, where t is the number of years since 1990.

W(t) = 0.014t2 – 0.08t + 7.96M(t) = 0.012t2 – 0.12t + 6.65



1. Find an equation of the sum function W + M.2. Perform a unit analysis of the expression

W(t) + M(t).3. Find (W + M)(27). What does it mean in this

situation?4. Find an equation of the difference function W – M.5. Find (W – M)(27). What does it mean in this

situation?


Solution

1.

2. For the expressions W(t) + M(t), we have

The units of the expression are millions of students.

( ( )))( () WW t tM M t 220.014 0.08 7. 0.012 0.12 6.69 56 tt t t

20.026 0.20 14.61t t


Solution

3.

(W + M)(27) = 0.026(27)2 – 0.20(27) + 14.61 ≈ 28.16

This means the total enrollment for women and men in 1990 + 27 = 2017 will be about 28.2 millions students, according to the model.


Solution4.

5. (W – M)(27) = 0.002(27)2 + 0.04(27) + 1.31 ≈ 3.85

This means in 2017 women’s enrollment will exceed men’s enrollment by about 3.9 million students, according to the model.

( ( )))( () WW t tM M t

220.014 0.08 7.9 0.012 0.1 6 6516 2 .tt t t 2 27.96 6.0.014 0.012 60.08 0.12 5t tt t 2 0.00.0 1 3102 4 .tt

copyright © 2015, 2008, 2011 pearson education, inc. section 6.1, slide 1 chapter 6 polynomial...

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