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Period: _________________________
Unit 6B
Polynomials: Solving Higher Degree
Algebra II
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Video
Section 6.6
Section 6.7
Section 6.8
Section 6.9
Section 6.10
Classwork
Section 6.6
Section 6.7
Section 6.8
Section 6.9
Section 6.10
Section 6.10
Worksheet
Practice Test
Unit 6B Exam
Due
Section 6.6
Section 6.7
Section 6.8
Section 6.9
Section 6.10
Practice Test
Section 6.6 Evaluating Polynomial Functions
Objective(s): Use synthetic substitution to evaluate polynomials.
Essential Question:
Homework: Assignment 6.6. #1 – 12 in the homework packet.
Notes:
Types of polynomials
Leading
Coefficient
Degree
Type
Example
0
1
2
3
4
Constant
Linear
Quadratic
Cubic
Quartic
f(x) = 4
f(x) = 2x + 1
f(x) = -x2 + 7x - 5
f(x) = 6x3 + x2 - 19
f(x) = -7x4 + 2x2 + 3
2
-1
6
-7
Depending on the polynomial, it can be easier to evaluate (plug in a number) using a process called
synthetic substitution. Let’s suppose you had a polynomial f(x) = 2x4 + x2 – 12x + 6 and you wanted to
evaluate it for x = 2. Traditionally, you would evaluate it in the following way.
f ( x)  2 x 4  x 2  12 x  6
f (2)  2(2) 4  (2) 2  12(2)  6
 2(16)  4  24  6
 32  4  24  6
 18
Sometimes it is quicker and easier to evaluate using synthetic substitution.
1. Rewrite the polynomial as f(x) = 2x4 + 0x3 + x2 – 12x + 6 (Hint: if the degree is 4, then you need
FIVE terms – one for each power of x plus the constant.)
2. Take all of the coefficients and write them in a row.
3. Leaving space for a second row, draw a giant L.
Reflection:
2 0 1 12 6
1
4. Write the 2 (evaluate for x = 2) on the outside of the giant L.
2 2 0 1 12 6
2 2 0 1 12 6
5. Drop the first number on the left of the top row.
2
6. Multiply this number by the number on the outside and put the result under the 0. 2 × 2 = 4
2 2 0 1 12 6
4
2
2 2 0 1 12 6
7. Now add the 0 and the 4. 0 + 4 = 4 and put this under the 0 and 4.
4
2 4
8. Multiply the 4 with the outside 2 and put the result under the 1. Then add.
2 2 0 1 12 6
4 8
2 4 9
2 2 0 1 12 6
9. Repeat for all of the numbers.
4 8
2 4 9
Notice how the last number in the bottom row is the same as f(2). This is NOT a coincidence.
Use synthetic substitution to evaluate.
Example 1:
Reflection:
If f(x) = -3x3+ 2x2 – 5, find f(4) (Are there any missing terms??? Remember to write a 0.)
2
Example 2:
If g(x) = -x5 – 4x3 + 6x2 – x, find g(-2)
Example 3:
If f(x) = x4 – 6x3 + 8x2, find f(3).
Example 4:
If h(x) = x5 – 3x3 + 1, find h(-1).
Example 5:
For f (x) = x³ - 3x² + 4x -2 show that f(1) = 0
1 1 3 4 2
This means that 1 is a ______________of the polynomial x³ - 3x² + 4x -2 .
x³ - 3x² + 4x -2 = 0 has ______zeros.
What is one of them ? _________
If x = _______is a solution, then what is a factor of the polynomial? _____________________
Reflection:
3
Section 6.7 The Remainder Theorem
Objective(s): Divide polynomials and relate the result to the remainder theorem and the factor
theorem.
Essential Question: If f(x) is a polynomial that has (x - 7) as a factor, what do you know about the value
of f(7)?
Homework: Assignment 6.7. #13 – 23 in the homework packet.
Notes:
If a polynomial f(x) is divided by x - k, then the remainder is r = f(k).
Let f ( x)  3x3  2 x 2  2 x  5
If we use long division to divide x – 2 into our polynomial, we have
x  2 3x3  2 x 2  2 x  5
3x 2
 x  2 3x3  2 x 2  2 x  5
divide 3x3 by x and get 3x2, put on top and multiply
3x 2
 x  2 3x3  2 x 2  2 x  5
3x  6 x
3
now subtract
2
What is your remainder? _________________
Now, what would happen if we were to use synthetic substitution?
2 3 2 2 5
Notice that the other numbers are the coefficients of the answer(quotient).
Reflection:
4
The remainder theorem says that using synthetic substitution for x = 2 will give you the same remainder
as dividing by x – 2.
Find the remainder.
Example 1:
(x3 – x2 + 5 ) ÷ (x + 2)
(Are there any missing terms??? Remember to write a 0.)
Divide using synthetic substitution.
Example 2:
(10x4 + 5x3 + 4x2 – 9 ) ÷ (x + 1)
Example 3:
(2x4 – 6x3 + x2 – 3x – 3 ) ÷ (x – 3)
Reflection:
5
Section 6.8 The Factor Theorem
Objective(s): Divide polynomials and relate the result to the remainder theorem and the factor
theorem.
Essential Question: If f(x) is a polynomial that has (x - 7) as a factor, what do you know about the value
of f(7)?
Homework: Assignment 6.8. #24 – 39 in the homework packet.
Notes:
How do you know if 23 is a factor of 1766?
The Factor Theorem states that a polynomial f(x) has a factor x - k if and only if f(k) = 0.
If the remainder is zero, then x - k is a factor.
In addition, k is a solution/root of the polynomial.
Use synthetic substitution to determine whether the binomial is a factor of f(x).
Example 1:
f(x) = x3 – 8x2 + 21x – 18; x – 2
Yes
No
Example 2:
f(x) = x3 + 9x2 + 14x – 24; x – 3
Yes
No
Using synthetic substitution, is the given number a root/zero of the polynomial?
Example 3:
Reflection:
Is 5 a zero (or root) of f(x) = x3 – 10x2 + 11x + 70?
Yes
No
6
Example 4:
Is 4 a zero (or root) of f(x) = x3 – 6x2 + 11x – 6?
Yes
No
Factor the polynomial f(x) given one of the zeros.
Example 5:
f(x) = x3 – 4x2 – 17x + 60 given 3 is a zero/root
Example 6:
f(x) = x3 + 3x2 – 16x – 48 given 4 is a zero/root
Given one zero/root, find the others.
Example 7:
f(x) = x3 – 12x2 + 44x – 48; x = 2
Example 8:
f(x) = x3 + 11x2 + 36x + 36; x = -2
Reflection:
7
Sample CCSD Common Exam Practice Question(s):
1. What is x 3  3 x 2  9 x  2 divided by x  2 ?
A.
x 2  x  11
B.
x 2  2 x  11
C.
x2  5x  1
D. x 2  4 x  5
Reflection:
8
Section 6.9 Using the Fundamental Theorem of Algebra
Objective(s): Use the Fundamental Theorem of Algebra to determine the number of zeros of a
polynomial function. Solve polynomial equations.
Essential Question: What is the conjugate of a complex number, and why is it important when finding all
of the zeros of a polynomial function?
Homework: Assignment 6.9. #40 – 46 in the homework packet.
Notes:
The Fundamental Theorem of Algebra states that the degree of the polynomial is the same as the TOTAL
number of real and imaginary solutions/roots/zeros.
For the function, find the number of zeros that the function has.
Example 1:
f(x) = x5 – 5x3 + 10x + 4
Number of zeros = ______________
Example 2:
f(x) = x8 – 7x4 + 2x3 + 4x2 – 12x – 7
Number of zeros = ______________
The complex zeros of a polynomial function with real coefficients always occur in complex conjugate
pairs. That is, if a + bi is a zero, then a – bi must also be a zero.
Find the missing root.
5 are roots of f(x) = x4 – x3 – 11x2 + 5x + 30, what is the missing root?
Example 3:
If -2, 3, and
Example 4:
If -4 and 2 + 3i are roots of f(x) = x3 – 3x + 52, what is the missing root?
Example 5:
If 2i and 3i are roots of f(x) = x4 + 13x2 + 36, what are the missing roots?
Example 6:
If 7 is a zero of a polynomial, what is the factor?
Example 7:
If -2, 3, and 5 are zeros of a polynomial, what are the factors?
Find a polynomial function with -2, 3, and 5 as zeros. (Hint: multiply it out)
Reflection:
9
Example 8:
Find a polynomial function with -1, 2i, and -2i as zeros.
Example 9:
Find a polynomial function with 3, 4i, and -4i as zeros.
Sample CCSD Common Exam Practice Question(s):
1. According to the Fundamental Theorem of Algebra, how many complex zeros does the polynomial
f  x   5 x 4  2 x 3  x  1 have?
A.
B.
C.
D.
2
3
4
5
Reflection:
10
Section 6.10 Finding Rational Zeros
Objective(s): Identify all rational zeros of a polynomial function by using the rational root theorem. Find
rational zeros of a polynomial.
Essential Question: If the leading coefficient of a polynomial with integer coefficients is 1, what type of
numbers must any possible rational zeros be?
Homework: Assignment 6.10. #47 – 58 in the homework packet.
Notes:
Rational Zero Theorem – If a polynomial has integer coefficients, then every rational zero of the
polynomial has the following form
all factors of constant
all factors of lead coefficient
For example, all the possible rational zeros of f(x) = x3 + 2x2 – 11x – 12 would be
1, 2, 3, 4, 6, 12
which would simplify to 1,  2,  3,  4,  6,  12 .
1
All the possible rational zeros of f(x) = 2x3 + 7x – 12 would be
1 3
1, 2, 3, 4, 6, 12
which would simplify to 1,  ,  , 2,  3,  4,  6,  12 .
2 2
1, 2
List all possible rational zeros of f.
Example 1:
f(x) = 2x5 + 3x3 – 7
Example 2:
f(x) = 2x4 + 8x3 – 7x2 – 9x + 8
Reflection:
11
Example 3:
f(x) = 3x5 + 12x4 + 21x3 – 8x2 – x – 27
Example 4:
Which is NOT a possible rational root of f(x) = 6x4 + 7x2 – 3x + 2
a) -2
b) 2/3
c) ½
d) -3
Procedure to find 3 roots
1. List all possible rational roots
2. Use synthetic substitution until you find a root (remainder = 0)
3. Use the new coefficients and find the remaining 2 roots by factoring or the quadratic formula.
x
b  b2  4ac
2a
Given the polynomial function f(x), find all real and complex zeros.
Example 5:
Reflection:
f(x) = x3 – 7x + 6
12
Given the polynomial function f(x), find all real and complex zeros.
Example 6:
f(x) = x3 – 4x2 + 9x – 10
For the polynomial function, solve for all its zeros (both real and imaginary), and graph it - include a
description of its degree, roots, and end behavior.
Example 7:
f(x) = – x3 – 3x2 + 13x + 15
Degree = _______
Possible roots = _________________________
Actual roots = __________________________
As x → + ∞ f(x) → ____________
As x → - ∞ f(x) → ____________
Reflection:
13
Example 8:
f(x) = 3x3 + 7x2 – 7x – 3
Degree = _______
Possible roots = _________________________
Actual roots = __________________________
As x → + ∞ f(x) → ____________
As x → - ∞ f(x) → ____________
Reflection:
14
Example 9:
f(x) = x4 + 2x3 – 9x2 – 2x + 8
Degree = _______
Possible roots = _________________________
Actual roots = __________________________
As x → + ∞ f(x) → ____________
As x → - ∞ f(x) → ____________
Reflection:
15