7.3.1 Products and Factors of Polynomials 7.3.1 Products and Factors of Polynomials Objectives: Multiply and factor polynomials Use the Factor Theorem

7.3.1 Products and Factors of 7.3.1 Products and Factors of PolynomialsPolynomials


Objectives: •Multiply and factor polynomials•Use the Factor Theorem to solve problems

Real-World Application

Objective: Multiply and factor polynomials

Real-World Application


If I wanted to maximize the volume of this open-top box, what do you hypothesize I would need to do? In other words, what important information do I need to find?

Collins Type 1


Example 1Write the function f(x) = (x – 1)(x + 4)(x – 3) as a polynomial function in standard form.

(x – 1)(x + 4)(x – 3)

= (x – 1)= (x – 1)= x(x2 + x – 12)

[(x + 4)(x – 3)](x2 + x – 12) – 1(x2 + x –

12)= x3 + x2

– 12x

– x2 – x + 12= x3 – 13x +

12

f(x) = x3 – 13x + 12


Example 2Factor each polynomial.a) x3 – 16x2 +

64xx x x

= x(x2 – 16x + 64)= x(x – 8)(x – 8)

b) x3 + 6x2 – 5x - 30

= (x3 + 6x2) + (-5x – 30)= x2(x + 6) – 5(x + 6)= (x + 6)(x2 – 5)

(x + 6)

(x + 6)


Factoring the Sum and Difference of Two Cubes

a3 + b3 =

a3 - b3 =

(a + b)(a2 – ab + b2)(a - b)(a2 + ab + b2)


Example 3Factor each polynomial.a) x3 +

125

b) x3 - 27

= x3 + 53= (x + 5)(x2 – 5x + 25)

= x3 - 33= (x - 3)(x2 + 3x + 9)


Factor Theoremx – r is a factor of the polynomial expression that defines the function P iff r is a solution of P(x) = 0, that is, iff P(r) = 0.

Objective: Use the Factor Theorem to solve problems

Example 4Use substitution to determine whether x – 1 is a factor of x3 – x2 – 5x – 3.

Let x3 – x2 – 5x – 3 = 0

f(1) = (1)3 – (1)2 – 5(1) - 3f(1) = 1 – 1 – 5 - 3f(1) = -8Since f(1) does not equal zero, x – 1 is not a factor.


Practice1) Factor each polynomial.

2) Use substitution to determine whether x + 3 is a factor of x3 – 3x2 – 6x + 8.

x3 + 1000

x3 - 125


Collins Type 2

If p(-2) = 0, what does that tell you about the graph of p(x)?


Homework

Lesson 7.3 Exercises 51-69 odd



Objectives: •Divide one polynomial by another synthetic division•Divide one polynomial by another using long division

Example 1Use synthetic division to find the quotient: (6 – 3x2 + x + x3) ÷ (x – 3)

13 -3 1 6

Are the conditions for synthetic division met?

Objective: Divide one polynomial by another using synthetic division

Step 1: Write the opposite of the constant of the divisor on the shelf, and the coefficients of the dividend (in order) on the right.

Example 1Use synthetic division to find the quotient: (x3 – 3x2 + x + 6) ÷ (x – 3)

13

1

-3 1 6

Step 2: Bring down the first coefficient under the line.



Step 3: Multiply the number on the shelf, 3, by the number below the line and write the product below the next coefficient.

13

13

-3 1 6



Step 4: Write the sum of -3 and 3 below the line.

13

130

-3 1 6



Repeat steps 3 and 4.

13

130

01

-3 1 6



Repeat steps 3 and 4.

13

130

01

39

-3 1 6


Example 1

(x3 – 3x2 + x + 6) ÷ (x – 3)

13

130

01

39

-3 1 6

The remainder is 9 and the resulting numbers are the coefficients of the quotient.

x2 + 1 +

x – 39

Use synthetic division to find the quotient:


Remainder

Answer:

PracticeGroup 1 & 5:

Divide: (x3 + 3x2 – 13x - 15) ÷ (x – 3)


Group 2 & 6:

Divide: (x3 - 2x2 – 22x + 40) ÷ (x – 4)

Group 3 & 7:

Divide: (x3 - 27) ÷ (x – 3)

Group 4 & 8:

Divide: (x5 + 6x3 - 5x4 + 5x - 15) ÷ (x – 3)

Do you remember long division?

Using long division: 745 ÷ 3

7453248

6

14

1 2

2524

1

-

-

-

Answer: 248 1

3

Objective: Divide one polynomial by another using long division

(–14x + 56)

x – 4 x3 – 2x2 – 22x + 40

x2

(x3 – 4x2)

2x2

+ 2x

(2x2 – 8x) –14x

– 14

– 16

x2 + 2x – 14 – x –

4

16

Example 2Using long division: (x3 – 2x2 – 22x + 40) ÷ (x – 4)

x – 4

- 16


-

-

–

– 22x

+ 40

Answer:

Example 3Use long division to determine if x2 + 3x + 2 is a factor of x3 + 6x2 + 11x + 6.

x2 + 3x + 2

x3 + 6x2 + 11x + 6 (x3 + 3x2 +

2x ) 3x2 + 9x

x

(3x2 + 9x + 6)

+ 3

0x2 + 3x + 2 is a

factor because the remainder is 0

+ 6


-

-

Practice

Group 4 & 8:

Divide: (x3 + 3x2 – 13x - 15) ÷ (x2 – 2x – 3)


Group 3 & 7:

Divide: (x3 + 6x2 – x - 30) ÷ (x2 + 8x + 15)

Group 2 & 6:

Divide: (10x - 5x2 + x3 - 24) ÷ (x2 – x + 6)

Group 1 & 5:

Divide: (x3 - 8) ÷ (x2 – 2x + 4)

Collins Type 1

When dividing x3 + 11x2 + 39x + 45 by x + 5, would you use synthetic division or long division? Explain why.

Objective: Divide one polynomial by another

Homework

Lesson 7.3 Read Textbook Pages 442-444 Exercises 71-89 odd

Example 3Given that 2 is a zero of P(x) = x3 – 3x2 + 4, use division to factor x3 – 3x2 + 4.Since 2 is a zero, x = 2

, so x – 2 = 0

, which means x – 2is a factor of x3 – 3x2 + 4. (x3 – 3x2 + 4) ÷ (x –

2) Method 1 Method 2

- (–2x + 4)

x – 2 x3 – 3x2 + 0x + 4

x2

- (x3 – 2x2)

-x2 + 0x

- x

- (-x2 + 2x) –2x +

4

– 2

0

1

2

-1

-2

-2

-4

0

2 1 -3 0 4

x3 – 3x2 + 4 = (x – 2)(x2 – x – 2)


PracticeGiven that -3 is a zero of P(x) = x3 – 13x - 12, use division to factor x3 – 13x – 12.


Groups 1-4 use Method 1 (Long Division)

Groups 5-8 use Method 2 (Synthetic Division)

Remainder TheoremIf the polynomial expression that defines the function of P is divided by x – a, then the remainder is the number P(a).

Objective: Use the Remainder Theorem to solve problems

Example 6Given P(x) = 3x3 – 4x2 + 9x + 5 is divided by x – 6, find the remainder.

3

1814

84

93

558

563

6 3 -4 9 5

Method 1 Method 2

P(6) = 3(6)3 – 4(6)2 + 9(6) + 5 = 3(216) – 4(36) + 54

+ 5= 648 – 144 + 54 + 5

= 563


PracticeGiven P(x) = 3x3 + 2x2 + 3x + 1 is divided by x + 2, find the remainder.


A company manufactures cardboard boxes in the following way: they begin with 12"-by-18" pieces of cardboard, cut an x"-by-x" square from each of the four corners, then fold up the four flaps to make an open-top box.

a. Sketch a picture or pictures of the manufacturing process described above. Label all segments in your diagram with their lengths (these will be formulas in terms of x).

b. What are the length, width, and height of the box, in terms of x?

c. Write a function V(x) expressing the volume of the box.

d. Only some values of x would be meaningful in this problem. What is the interval of appropriate x-values?

e. Using the interval you just named, make the graph V(x) on your calculator, then sketch it on paper.

f. What value of x would produce a box with maximum volume?

g. What are the dimensions and the volume for the box of maximum volume?

Documents

7.3.1 Products and Factors of Polynomials 7.3.1 Products and Factors of Polynomials Objectives: Multiply and factor polynomials Use the Factor Theorem