Download Chapter 3 Study Guide

Chapter 3 Study Guide
3.1 Polynomials
The degree of a polynomial is the value of the exponent of the term of the greatest degree.
A polynomial is in standard form when the terms are arranged in order with exponents from greatest to
least.
To arrange the polynomial 3x 2  x 4  2x  6x 5  7 in standard form, order the terms greatest to least
exponent
Degree
Polynomial in Standard Form
0
8
1
2x  3
2
x  4x  5
3
4x 3  x
4
6x 4  x 3  5x 2  3x  1
5
9x 5  x 3  1
Constants have degree 0.
This third degree polynomial
has 2 terms.
2
6 is the leading
coefficient of this
polynomial.
This fifth degree polynomial
has 3 terms.
6x 5  x 4  3x 2  2x  7
To add polynomials:
− Write each polynomial in standard form.
− Align like terms vertically.
− Add like terms.

 

Add: 6 x  2 x 3  5 x 2  1  4 x 2  2 x  x 3 .
2x  5x  6x  1
 x 3  4x 2  2x
3
2
3x 3  x 2  8x  1
To subtract polynomials, add the opposite vertically.

 

Subtract: 6 x  2 x 3  5 x 2  1  4 x 2  2 x  x 3 .

 

Add the opposite: 6 x  2 x 3  5 x 2  1  4 x 2  2 x  x 3 .
2x 3  5x 2  6x  1
 (x 3  4x 2  2x)
Add like terms vertically.
x 3  9x 2  4x  1
Write each polynomial in standard form. Add or subtract.

 

1. 3 x 2  2 x 3  x  6 x  2 x 2  1

 

2. 6 x 2  4 x  1  2 x  x 2  1
3.2 Multiplying Polynomials
Use the Distributive Property to multiply two polynomials.
Distribute each term of the first polynomial to each term of the second polynomial.
Multiply: (x  2)(4x 2  3x  1).
Horizontal Method: (x  2)(4x 2  3x  1)
[x(4x 2)  x(3x)  x(1)]  [2(4x 2)  2(3x)  2(1)]
4x 3  3x 2  x  8x 2  6x  2
Multiply.
4x 3  3x 2  8x 2  x  6x  2
Group like terms.
4x 3  5x 2  7x  2
Combine like terms.
3. (x  3)(x 2  2x  1)
4.
(x  3) 3
3.3 Dividing Polynomials
Divide: (6x 2  x  8)  (2x  1).
Step 1 Divide the first term of the dividend, 6x 2, by the first term of the divisor, 2x.
3x
2x  1 6 x  x  8
2
Divide: 6x 2  2x  3x.
 (6x 2  3x)
4x  8
Multiply the complete divisor: 3x(2x  1)  6x 2  3x.
Subtract and bring down.
Remember to use the
Step 2 Divide the first term of the difference, 4x, by the
Distributive Property
first term of the divisor, 2x.
when you subtract.
3x  2
2x  1 6 x 2  x  8
 (6x 2  3x)
4x  8
 (4x  2)
10
Multiply: 3x(2x  1)  6x 2  3x.
Divide: 4x  2x  2.
Multiply the complete divisor: 2(2x  1)  4x  2.
Subtract. Use the Distributive Property.
Step 3 Write the quotient including the remainder.
 6x
2

 x  8   2x  1  3 x  2 
10
2x  1
When the divisor is in the form (x  a), use synthetic division to divide.
Divide: (2x 2  x  10)  (x  3).
Step 1 Find a. The divisor is (x  3). So, a  3.
Step 2 Write a in the upper left corner.
Then write the coefficients of the dividend.
3
2 1 10
2, 1, and 10 are the coefficients of 2x 2  x  10.
Step 3 Draw a horizontal line. Copy the first coefficient below the line.
3 2 1 10
2
Step 4 Multiply the first coefficient by a, or 3. Write the product in the second column. Add the numbers in
the column.
3
2 1 10
6
2 5
2a  2(3)  6
Step 5 Multiply that sum by a, or 3. Write the product in the third column.
Add the numbers in the column.
Draw a box around the last number. It is the remainder.
3
2 1 10
6
15
2 5
5
Step 6 Write the quotient.
5. x  5 3 x 2  5 x  50
5a  5(3)  15
The numbers
5 in the bottom row are the
2x  5  of the quotient.
coefficients
x 3
6. 3 x  2 6 x 2  7 x  6
Use synthetic division to divide.
7.(4x 2  7x  10)  (x  2)
8. (2x 2  6x  12)  (x  5)
3.4 Factoring Polynomials
Sometimes you can use grouping to factor a third degree polynomial. To factor by grouping means to group
terms with common factors. Then factor the common factors. Continue to factor until the expression can no
longer be factored.
Factor: x 3  4x 2  9x  36.
Start by grouping terms to factor out the greatest possible power of x.
x 3  4x 2  9x  36
9 is a factor of 9 and 36.
x 2 is a factor of
3
2
2
(x

4x
)

(9x

36)
x and 4x .
(x  4) is a common factor.
x 2(x  4)  9(x  4)
(x 2  9) is the difference of squares.
(x  4) (x 2  9)
(x  4) (x  3) (x  3)
Recall that (a 2  b 2)  (a  b)
(a  b). So (x 2  9)
 (x  3) (x  3).
Use special rules to factor the sum or difference of two cubes.
Recognizing these common cubes can help you factor the sum or difference of cubes.
1 3  1, 2 3  8, 3 3  27, 4 3  64, 5 3  125, and 6 3  216
Rule for the Sum of Two Cubes: a 3  b 3  (a  b)(a 2  ab  b 2).
Factor: y 3  64.
y 3  64
Identify the cubes: y 3 and 64  4 3.
y3  43
Write the expression as the sum of two cubes.
(y + 4)(y 2  4y  16)
Use the rule to factor.
Rule for the Difference of Two Cubes: a 3  b 3  (a  b) (a 2  ab  b 2).
Factor: 8x 3  125.
8x 3  125
Identify the cubes: (2x) 3 and 125  5 3.
(2x) 3  5 3
Write the expression as the difference of two cubes.
(2x  5)(4x  10x  25)
2
Use the rule to factor.
Using the rule: a  2x and b  5.
So a 2  (2x) 2  4x 2, ab  (2x)(5)  10x, and b 2  25.
Factor each expression.
9. x 3  3x 2  4x  12
11. y 3  27
10. x 3  6x 2  x  6
12. x 3  1
3.5 Finding Real Roots of Polynomial Equations
To find the roots of a polynomial equation, set the equation equal to zero. Factor the polynomial expression
completely. Then set each factor equal to zero to solve for the variable.
Solve the equation: 2x 5  6x 4  8x 3.
Step 1
To set the equation equal to 0, rearrange the equation so that all the terms are on one side.
2x 5  6x 4  8x 3
2x 5  6x 4  8x 3  0
Step 2
Look for the greatest number and the greatest power of x that can be factored from each term.
2x 5  6x 4  8x 3  0
The GCF is 2x 3.
2x 3(x 2  3x  4)  0
Step 3
Factor the quadratic.
2x 3(x 2  3x  4)  0
2x 3(x  4) (x  1)  0
Step 4
Set each factor equal to 0.
2x 3  0
Step 5
x40
x10
Solve each equation.
2x 3  0
x40
x0
x  4
x10
x1
The solutions of the equation are called the roots.
The roots are 4, 0, and 1.
13. 2x 3  6x 2  36x  0
14. 2x 6  32x 4  0
3.6 Fundamental Theorem of Algebra
Write the simplest polynomial function with roots 4, 2, and 3.
Step 1
Write the factors of the polynomial, P(x)  0.
(x  4)(x  2)(x  3)  0
Step 2
Multiply the first two factors, (x  4)(x  2).
(x 2  6x  8)(x  3)  0
Step 3
Multiply (x 2  6x  8)(x  3). Then simplify.
Root (a)
4
2
3
Factor
(x  a)
x4
x2
x3
x 3  3x 2  6x 2  18x  8x  24  0
x 3  3x 2  10x  24  0
The function is P(x)  x 3  3x 2  10x  24  0.
To solve x 4  x 3  5x 2  x  6  0 means to find all the roots
of the equation. A fourth degree equation has 4 roots.
Step 1
Identify possible real roots.
Possible roots,: 1, 2, 3, 6
Graph y  x 4  x 3  5x 2  x  6.
Step 2
Step 3
Test 2 as a root using synthetic substitution.
2 1 1 5 1 6
2 6 2 6
1 3 1 3 0
Test 3 as a root using synthetic substitution.
3 1 3 1 3
3 0 3
1 0 1 0
Step 4
Find the remaining roots.
The remainder is 0, so 2 is a root.
(x  2)(x 3  3x 2  x  3)  0
The remainder is 0, so 3 is a root.
(x  2) (x  3) (x 2  1)  0
x2  1  0
xi
The roots of the equation are 2, 3, i, and i.
Write the simplest polynomial function with the given roots.
15. 5, 1, and 2
Solve the equation by finding all roots.
16.
x 3  6x 2  2x  12  0
3.7 Investigating Graphs of Polynomial Functions
Examine the sign and the exponent of the leading term (term of greatest degree) of a polynomial P(x) to
determine the end behavior of the function.
Even degree functions: Exponent of leading term is even.
Read:
Positive leading coefficient
Negative leading coefficient
As
x
approaches
As x  , P(x)  .
As x  , P(x)  .
positive infinity, P(x)
As x  , P(x)  .
As x  , P(x)  .
approaches negative
infinity.
Example: P(x)  3x 4  2x 3  5
End behavior:
Leading term: 3x 4
Sign: positive
Degree: 4, even
As x  , P(x)  .
As x  , P(x)  .
Odd degree functions: Exponent of leading term is odd.
Positive leading coefficient
As x  , P(x)  .
As x  , P(x)  .
Negative leading coefficient
As x  , P(x)  .
As x  , P(x)  .
Example: P(x)  2x 5  6x 2  x Leading term: 2x 5
Sign: negative
Degree: 5, odd
Fill in the table below:
Polynomial
17
P(x)  x 2  3x  6
Leading
Coefficient
Degree
End Behavior
As x  , P(x)  
As x  , P(x)  
18.
P(x)  3x 3  2x  5
As x  , P(x)  ____
As x  , P(x)  ____
3.8 Transforming Polynomial Functions
Write the rule for function g(x).
19. g(x) reflects f(x)  x 3  2x  3 across the x-axis and translate 1 unit up
20. g(x) reflects f(x)  x 2  x  1 across the y-axis and translate 2 units left
3.9 Curve Fitting with Polynomial Models
Finite Differences
Function Type
Linear
Quadratic
Cubic
Quartic
Quintic
1
2
3
4
5
First
Second
Third
Fourth
Fifth
Degree
Constant Finite
Differences
x
10
20
30
40
50
y
2633
3812
4862
6529
9552
First
Differences
3812  2633
4862  3812
6529  4862
9552  6529
1179
1050
1667
3023
Second
Differences
Third
Differences
1050  1179
1667  1050
3023  1667
129
617
1356
617  (129)
1356  617
746
739
Since the third differences are reasonably close, you can
use a cubic function to model the data.
Use the cubic regression feature on your calculator.
Use the coefficients a, b, c, and d to write the function.
f(x)  0.12x 3  8.06x 2  273.1x  584.6
Use finite differences to determine the degree of the polynomial that best describes
the data.
21.
x
0
1
2
3
4
5
y
5
1
12
29
53
85
a. Which differences are constant?
b. Identify the degree of the polynomial of best fit.
c. Write the function
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