Chapter 6

Polynomial Functions and Inequalities

6.1 Properties of Exponents

Negative Exponents• For any real number a = 0 and any integer n,

a-n =

– Move the base with the negative exponent to the other part of the fraction to make it positive

na

1

Product of Powers

• am · an =

*Add exponents when you have multiplication with the same base

Quotient of Powers• For any real number a = 0,

= am – n

*subtract exponents when you have division with the same base

n

m

a

a

am+n

Power of a Power• (am)n =

*Multiply exponents when you have a power to a power

Power of a Product(ab)m=

*Distribute the exponents when you have a multiplication problem to a power

amn

ambm

Power of a Quotient

* distribute the exponent to both numerator and denominator, then use other property rules to simplify

Zero Powera0 =

* any number with the exponent zero = 1

m

b

a

n

b

a

1

m

m

b

an

n

a

b

n

n

b

a

Examples

• 1. 52 ∙ 56

• 2. -3y ∙ -9y4

• 3. (4x3y2)(-5y3x)

• 4. 597,6520

5. (29m)0 + 70

6. 54a7b10c15

-18a2b6c5

7. (3yz5)3

8. 8r4

2r-4

9. y-7y 13. 30 – 3t0

y8

10. 4x0 + 5 14. (2-2 x 4-1)3

11. x-7y-2

x2y2

12. (6xy2)-1

6.2 Operations with Polynomials

• Polynomial: A monomial or a sum of monomials

– Remember a monomial is a number, a variable, or the product of a number and one or more variables

Rules for polynomials

• Degree of a polynomial = the highest exponent of all the monomial terms

• Adding and Subtracting = combine like terms

• Multiplying = Distribute or FOIL

Examples

1. (3x2-x+2) + (x2+4x-9)

2. (9r2+6r+16) – (8r2+7r+10)

3. (p+6)(p-9)

4. 4a(3a2+b)

5. (3b-c)2

6. (x2+xy+y2)(x-y)

6.3 Dividing Polynomials

2

3232

3

12159

xy

xyxyyx 1.

When dividing by a monomial:Divide each term by the denominator separately

2.

ab

baabba

5

10155 4332

Dividing by a polynomial

• Long Division: rewrite it as a long division problem

51522 xxx

1525 2 xxx

1. 2.4642 23 xxxx

Dividing by a polynomial

• Synthetic Division– Step 1: Write the terms of the dividend so that the degrees

of the terms are in descending order. Then write just the coefficients.

– Step 2: Write the constant r of the divisor x – r to the left. Bring the first coefficient down.

– Step 3: Multiply the first coefficient by r . Write the product under the second coefficient. Then add the product and the second coefficient.

– Step 4: Multiply the sum by r. Write the product under the next coefficient and add. Repeat until finished.

– Step 5: rewrite the coefficient answers with appropriate x values

ex 1: Use synthetic division to find (x3 – 4x2 + 6x – 4) ÷ (x – 2).

• Step 1

• Step 2

• Step 3

x3 – 4x2 + 6x – 4 1 - 4 6 - 4

1 – 4 6 – 4

1

1 - 4 6 - 4

1

2

- 2

- 4

2

- 4

- 8

x2 – 2x + 2 – 8

x - 5

Use synthetic division to solve each problem

2

81213 23

x

xxx 124 12448 xxxx2. 3.

6.4 Polynomial Functions

• Polynomial in one variable : A polynomial with only one variable

• Leading coefficient: the coefficient of the term with the highest degree in a polynomial in one variable

• Polynomial Function: A polynomial equation where the y is replaced by f(x)

State the degree and leading cofficient of each polynomial, if it is not a polynomial in

one variable explain why.1. 7x4 + 5x2 + x – 9

2. 8x2 + 3xy – 2y2

3. 7x6 – 4x3 + x-1

4. ½ x2 + 2x3 – x5

Evaluating Functions

• Evaluate f(x) = 3x2 – 3x +1 when x = 3

• Find f(b2) if f(x) = 2x2 + 3x – 1

• Find 2g(c+2) + 3g(2c) if g(x) = x2 - 4

End Behavior

• Describes the behavior of the graph f(x) as x approaches positive infinity or negative infinity.

• Symbol for infinity

End behavior Practice

f(x) - as x +

f(x) - as x -

End Behavior Practice

f(x) + as x +

f(x) - as x -

End Behavior Practice

f(x) + as x +

f(x) + as x -

The Rules in General

To determine if a function is even or odd

• Even functions: arrows go the same direction• Odd functions: arrows go opposite directions

To determine if the leading coefficient is positive or negative

• If the graph goes down to the right the leading coefficient is negative

• If the graph goes up to the right then the leading coeffiecient is positive

The number of zeros • zeros are the same as roots: where the graph crosses

the x-axis– The number of zeros of a function can be equal to the exponent

or can be less than that by a multiple of 2.• Example a quintic function, exponent 5, can have 5, 3 or 1 zeros

• To find the zeros you factor the polynomial

Critical Points• points where the graph changes direction.

– These points give us maximum and minimum values• Relative Max/Min

Put it all together

• For the graph given – Describe the end

behavior– Determine whether it is an

even or an odd degree– Determine if the leading

coefficient is positive or negative

– State the number of zeros

Cont…

• For the graph given – Describe the end behavior– Determine whether it is an

even or an odd degree– Determine if the leading

coefficient is positive or negative

– State the number of zeros

• For the graph given – Describe the end

behavior– Determine whether

it is an even or an odd degree

– Determine if the leading coefficient is positive or negative

– State the number of zeros

• For the graph given – Describe the end

behavior– Determine whether it is

an even or an odd degree

– Determine if the leading coefficient is positive or negative

– State the number of zeros

6.5 Analyze Graphs of Polynomial Functions

• Location Principle: used to find the numbers between which you find the roots/zeros

– Make a table to sketch the graph– Estimate and list the location of all the real

zeros • Zeros are between # and #

• Relative Maximum and Minimum:the y-coordinate values at each turning point in the graph of a polynomial.

*These are the highest and lowest points in the near by area of the graph

At most each polynomial has one less turning point than the degree

Find the location of all possible real zeros. Then name the relative minima and maxima as well as

where they occur

Ex 1. f(x) = x3 – x2 – 4x + 4

Find the location of all possible real zeros. Then name the relative minima and maxima as well as

where they occur

Ex 2. f(x) = x4 – 7x2 + x + 5

6.9 Rational Zero Theorem

Parts of a polynomial function f(x) o Factors of the leading coefficient = qo Factors of the constant = po Possible rational roots =

q

p

Ex 1: List all the possible rational zeros for the given function

a. f(x) = 2x3 – 11x2 + 12x + 9 b. f(x) = x3 - 9x2 – x +105

Finding Zeros of a function

• After you find all the possible rational zeros use guess and check along with synthetic division to find a number that gives you a remainder of 0!

• Then factor and or use the quadratic formula with the remaining polynomial to find any other possible zeros

Find all the zeros of the given function

• f(x) = 2x4 - 5x3 + 20x2 - 45x + 18

Ex 2: Find all the zeros of the given function

f(x) = 9x4 + 5x2 - 4

The volume of a rectangular solid is 1001 in3. The height of the box is x – 3 in. The

width is 4 in more than the height, and the lengthis 6 in more than the height. Find the dimensions

of the solid.

The volume of a rectangular solid is 675 cm3. Thewidth is 4 cm less than the height, and the lengthis 6cm more than the height. Find the dimensions

of the solid.