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7.3.1 Products and Factors of 7.3.1 Products and Factors of PolynomialsPolynomials
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
Objective: Multiply and factor polynomials
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
Objective: Multiply and factor polynomials
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
Objective: Multiply and factor polynomials
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)
Objective: Multiply and factor polynomials
Factoring the Sum and Difference of Two Cubes
a3 + b3 =
a3 - b3 =
(a + b)(a2 – ab + b2)(a - b)(a2 + ab + b2)
Objective: Multiply and factor polynomials
Example 3Factor each polynomial.a) x3 +
125
b) x3 - 27
= x3 + 53= (x + 5)(x2 – 5x + 25)
= x3 - 33= (x - 3)(x2 + 3x + 9)
Objective: Multiply and factor polynomials
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.
Objective: Use the Factor Theorem to solve problems
Practice1) Factor each polynomial.
2) Use substitution to determine whether x + 3 is a factor of x3 – 3x2 – 6x + 8.
x3 + 1000
x3 - 125
Objective: Use the Factor Theorem to solve problems
Collins Type 2
If p(-2) = 0, what does that tell you about the graph of p(x)?
Objective: Use the Factor Theorem to solve problems
Homework
Lesson 7.3 Exercises 51-69 odd
7.3.2 Products and Factors of 7.3.2 Products and Factors of PolynomialsPolynomials
7.3.2 Products and Factors of 7.3.2 Products and Factors of PolynomialsPolynomials
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.
Objective: Divide one polynomial by another using synthetic division
Example 1Use synthetic division to find the quotient: (x3 – 3x2 + x + 6) ÷ (x – 3)
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
Objective: Divide one polynomial by another using synthetic division
Example 1Use synthetic division to find the quotient: (x3 – 3x2 + x + 6) ÷ (x – 3)
Step 4: Write the sum of -3 and 3 below the line.
13
130
-3 1 6
Objective: Divide one polynomial by another using synthetic division
Example 1Use synthetic division to find the quotient: (x3 – 3x2 + x + 6) ÷ (x – 3)
Repeat steps 3 and 4.
13
130
01
-3 1 6
Objective: Divide one polynomial by another using synthetic division
Example 1Use synthetic division to find the quotient: (x3 – 3x2 + x + 6) ÷ (x – 3)
Repeat steps 3 and 4.
13
130
01
39
-3 1 6
Objective: Divide one polynomial by another using synthetic division
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:
Objective: Divide one polynomial by another using synthetic division
Remainder
Answer:
PracticeGroup 1 & 5:
Divide: (x3 + 3x2 – 13x - 15) ÷ (x – 3)
Objective: Divide one polynomial by another using synthetic division
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
Objective: Divide one polynomial by another using long division
-
-
–
– 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
Objective: Divide one polynomial by another using long division
-
-
Practice
Group 4 & 8:
Divide: (x3 + 3x2 – 13x - 15) ÷ (x2 – 2x – 3)
Objective: Divide one polynomial by another using long division
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)
Objective: Divide one polynomial by another
PracticeGiven that -3 is a zero of P(x) = x3 – 13x - 12, use division to factor x3 – 13x – 12.
Objective: Divide one polynomial by another
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
Objective: Use the Remainder Theorem to solve problems
PracticeGiven P(x) = 3x3 + 2x2 + 3x + 1 is divided by x + 2, find the remainder.
Objective: Use the Remainder Theorem to solve problems
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?
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