polynomials

Monomial: A number, a variable or the product of a number and one or more variables.

Polynomial: A monomial or a sum of monomials.

Binomial: A polynomial with exactly two terms.

Trinomial: A polynomial with exactly three terms.

Coefficient: A numerical factor in a term of an algebraic expression.

Degree of a monomial: The sum of the exponents of all of the variables in the monomial.

Degree of a polynomial in one variable: The largest exponent of that variable.

Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order.

What is the degree of the monomial?245 bx

The degree of a monomial is the sum of the exponents of the variables in the monomial.

The exponents of each variable are 4 and 2. 4+2 = 6.

The degree of the monomial is 6.

The monomial can be referred to as a sixth degree monomial.

A polynomial is a monomial or the sum of monomials

24x 83 3 x 1425 2 xx Each monomial in a polynomial is a term of the polynomial.

The number factor of a term is called the coefficient.

The coefficient of the first term in a polynomial is the lead coefficient.

A polynomial with two terms is called a binomial.

A polynomial with three terms is called a trinomial.

14 x

83 3 x

1425 2 xx

The degree of a polynomial in one variable is the largest exponent of that variable.

2 A constant has no variable. It is a 0 degree polynomial.

This is a 1st degree polynomial. 1st degree polynomials are linear.

This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.

This is a 3rd degree polynomial. 3rd degree polynomials are cubic.

Classify the polynomials by degree and number of terms.

Polynomial

a.

b.

c.

d.

5

42 x

xx 23

14 23 xx

DegreeClassify by

degreeClassify by

number of terms

Zero Constant Monomial

First Linear Binomial

Second Quadratic Binomial

Third Cubic Trinomial

To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term.

The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive.

745 24 xxx

x544x 2x 7

Write the polynomials in standard form.

243 5572 xxxx

32x4x 7x525x

)7552(1 234 xxxx

32x4x 7x525x

Remember: The lead coefficient should be

positive in standard form.

To do this, multiply the polynomial by –1 using

the distributive property.

Write the polynomials in standard form and identify the polynomial by degree and number of terms.

23 237 xx 1.

2. xx 231 2

23 237 xx

23 237 xx

33x 22x 7

7231 23 xx

723 23 xx

This is a 3rd degree, or cubic, trinomial.

xx 231 2 xx 231 2

23x x2 1

This is a 2nd degree, or quadratic, trinomial.

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Power Functions and Models Polynomial Functions and Models

14

15

Examples of Power Functions:

2

2

1)( xxh

16

The graph is symmetric with respect to the y-axis, so f is even.

The domain is the set of all real numbers. The range is the set of nonnegative numbers.

The graph always contains the points (-1,1), (0,0), and (1,1).

As the exponent increases in magnitude, the graph becomes more vertical when x <-1 or x >1, but for x near the origin the graph tends to flatten out and lie closer to the x-axis.

17

2 1 0 1 2

2

4

6

8

10

y x 4

y x 8

(1, 1)(-1, 1)

(0, 0)

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The graph is symmetric with respect to the origin, so f is odd.

The domain and range are the set of all real numbers.

The graph always contains the points (-1,-1), (0,0), and (1,1).

As the exponent increases in magnitude, the graph becomes more vertical when x > 1 or x <-1, but for x near the origin the graph tends to flatten out and lie closer to the x-axis.

19

2 1 0 1 2

10

6

2

2

6

10

y x 5

y x 9

(1, 1)(-1, -1)

(0, 0)

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A polynomial function is a function of the form

21

Polynomial. Degree 2.

Not a polynomial.

Not a polynomial.

Determine which of the following are polynomials. For those that are, state the degree.

(a)

(b)

(c)

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If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then

(a) r is an x-intercept of the graph of f.

(b) (x - r) is a factor of f.

23

Use the above to conclude that x = -1 and x = 4 are the real roots (zeroes) of f.

Math1414.8016 24

1 is a zero of multiplicity 2.-3 is a zero of multiplicity 1.-5 is a zero of multiplicity 5.

25

.

If r is a Zero or Odd Multiplicity

If r is a Zero or Even Multiplicity

26

Theorem

If f is a polynomial function of degree n, then f has at most n-1 turning points.

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TheoremFor large values of x, either positive or negative, the graph of the polynomial

resembles the graph of the power function.

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For the polynomial

(a) Find the x- and y-intercepts of the graph of f.

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

(c) Find the power function that the graph of f resembles for large values of x.

(d) Determine the maximum number of turning points on the graph of f.

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For the polynomial

(e) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis.

(f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

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(b) -4 is a zero of multiplicity 1. (crosses) -1 is a zero of multiplicity 2. (touches) 5 is a zero of multiplicity 1. (crosses)

(d) At most 3 turning points.

(a) The x-intercepts are -4, -1, and 5. y-intercept:

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Test number: -5

f (-5) 160

Graph of f: Above x-axis

Point on graph: (-5, 160)

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Test number: -2

f (-2) -14

Graph of f: Below x-axis

Point on graph: (-2, -14)

-4 < x <-1

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Test number: 0

f (0) -20

Graph of f: Below x-axis

Point on graph: (0, -20)

-1 < x < 5

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Test number: 6

f (6) 490

Graph of f: Above x-axis

Point on graph: (6, 490)

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8 6 4 2 0 2 4 6 8

300

100

100

300

500(6, 490)

(5, 0)(0, -20)

(-1, 0)

(-2, -14)(-4, 0)

(-5, 160)

Math1414.8016 36

An open box with a square base is to be made from a square piece of cardboard 30 inches wide on a side by cutting out a square from each corner and turning up the sides.

(a) Express the volume V of the box as a function of the length x of the side of the square cut from each corner.

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Volume = (length)(width)(height)

(b) Express the domain of V(x).

Domain of V(x) is determined by the fact that x has to be positive and 2x has to be less than 30.

Thus 0 < x < 15.

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V is largest (2000 cubic inches), when x = 5 inches.

(d) For what value of x is V largest?

(c) Graph V=V(x).

Fundamental Theorem of Algebra

Every complex polynomial function f (x) of degree n > 1 has at least one complex zero.

Fundamental Theorem of Algebra

Every complex polynomial function f (x) of degree n > 1 can be factored into n linear factors (not necessarily distinct) of the form

Find the zeros of

Use the zeros to factor f According to the quadratic formula

54)( 2 xxxf

ii

x

22

24

2

124

2

44

)1(2

)5)(1(4)4(4 2

)2)(2(

))2())(2(()(

ixix

ixixxf

Conjugate Pairs Theorem

Let f (x) be a complex polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the complex conjugate

Corollary

A complex polynomial f of odd degree with real coefficients has at least one real zero.

is also a zero of f.

Find a polynomial f of degree 4 whose coefficients are real numbers and that has zeros 1, 2, and 2+i.

f(x)