MATH 30-1
Polynomial Functions
Module Three Assignment
Module / Unit 3 - Assignment Booklet
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Math 30-1: Module 3 Lesson Assignment
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Math 30-1: Module 3 Lesson Assignment
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Lesson 1: Sketching Polynomial Functions
1. Describe the advantages of the factored form of a polynomial function and of the expanded
form of a polynomial function.
2. Without using technology, draw a possible sketch for each function provided and state the
following for each:
 end behaviour
 y-intercept
Be sure to justify your answers mathematically.
a. f(x)  3x2  5x  7
b. g(x)  5x3  2x2  2
Math 30-1: Module 3 Lesson Assignment
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c. h(x)  2x4  ax2  x  7, where a  R
3.
Without using technology, sketch the graphs for each function provided and state the
following for each:
Math 30-1: Module 3 Lesson Assignment




end behaviour
y-intercept
x-intercept(s)
multiplicity of each factor
Be sure to justify your answers mathematically.
a. f(x)  (x  2)(x  1)2
b. g(x)  2(x  2)3(x  1)
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Math 30-1: Module 3 Lesson Assignment
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c. h(x)  (x  a)(x  b)(x  c)2, where a  0, b  0, and c  0
4. A student was given a graphic design and drew a polynomial function to match the curve
(as shown in the image). Using your knowledge of polynomials, answer the following
questions.
adapted from Photodisc/Thinkstock
a. What is the least possible degree of the given polynomial function?
b. What is the sign of the leading coefficient of the given polynomial function?
LESSON 1 SUMMARY
Math 30-1: Module 3 Lesson Assignment
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In this lesson you learned how to sketch the graphs of polynomial functions. You learned that
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end behaviour can be determined by the degree and sign of the leading coefficient
x-intercepts can be determined by looking at the factors of the polynomial
the graph behaviour at the x-intercepts can be determined by examining the multiplicity of
the corresponding factors of the polynomial
the y-intercept can be determined by substituting zero for x
All functions of degree 3 or higher were shown in both expanded and factored forms. You will learn
how to factor these in the next lesson.
The number of times a factor is duplicated affects the graph’s behaviour around the corresponding xintercept.
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If a factor is repeated an odd number of times, the function changes sign (the graph
crosses the x-axis) at the corresponding x-intercept.
If a factor is repeated an even number of times, the function does not change sign (the
graph touches but does not cross the x-axis) at the corresponding x-intercept.
Consider the function c(x) = 0.8(x − 3)(x − 3)(x + 2). The factor x + 2 is repeated an odd number of
times, so the function changes sign at the corresponding x-intercept (x = −2). The factor x − 3 is
repeated an even number of times, so the function does not change signs at the corresponding xintercept (x = 3).
Mathematicians would say that the factor x − 3 has a multiplicity of 2 and that the factor x + 2 has a
multiplicity of 1.
Math 30-1: Module 3 Lesson Assignment
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Lesson 2: Factoring Polynomials
1. Completely factor each of the following polynomials.
a. P(x)  x3  4x2  x  6
b. P(x)  x4  4x3  3x2  14x  8
2. You are given f(x)  x4  3x3  kx  5. When f(x) is divided by x  3, the remainder is 166.
a. Determine the value of k.
b. Use your value of k from part a. to determine the remainder when f(x) is divided by x 
3.
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3. The polynomial shown at the beginning of the lesson could be written as
P(x)  x3  3x  x  3.
adapted from Stockbyte/Thinkstock
Is each of the following binomials a factor of the polynomial P(x)  x3  3x  x  3? Explain
your answers.
a. x  1
b. x  1
c. x  3
d. x  3
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LESSON 2 SUMMARY
The Remainder Theorem: The theorem states that when a polynomial is divided by the
binomial x − a (where a is a particular number), the remainder will be the number obtained when
a is substituted into the polynomial for x.
This means that when a polynomial in x, P(x), is divided by a binomial of the form x − a, the
remainder will be P(a).
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Division resulting in a zero remainder tells you that you have a factor.
The remainder when P(x) is divided by x − a is P(a).
The Factor Theorem: x − a is a factor of a polynomial in x, P(x), if and only if P(a) = 0.2
When looking for a factor of a polynomial, try values that are factors of the constant term of the
polynomial.
The Integral Zero Theorem. This theorem states that if x − a is a factor of a polynomial
function P(x) with integral coefficients, then a is a factor of the constant term of P(x).
Math 30-1: Module 3 Lesson Assignment
Lesson 3: Solving Polynomial Equations
1. Solve the following equations algebraically.
a. 2x3  3x2  11x  6  0
b. x3  x2  14x  10  14
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2. While solving an equation of the form ax3  bx2  cx  d  0 ( a, b, c, d Î R ), a student
created the following graph of f(x)  ax3  bx2  cx  d.
Describe how the graph can be used to determine the number of solutions on the interval
5  x  5. State the number of solutions on that interval.
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LESSON 3 SUMMARY
In this lesson you learned how to solve polynomial equations. In particular, you learned
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one side of the equation must be zero so the zero-product property can be used
synthetic division of polynomials
that there is a one-to-one correspondence among zeros of polynomials, xintercepts of graphs, and roots of equations
How to Factor Polynomials
Step 1: For any polynomial, f(x)
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the remainder theorem tells you that f(a) = the remainder when f(x) is divided by
x−a
the factor theorem tells you to look for values of a such that f(a) = 0
the integral zero theorem tells you to try factors of the constant term of f(x) for
values of a
Step 2: Use the factor theorem to determine one factor of f(x).
Step 3: Perform polynomial division to determine a second factor.
Step 4: This equivalent factored form can be written as f(x) = (x− a)Q(x) + R, where
Q(x) is the quotient and R is the remainder.
Step 5: Repeat until all factors have been identified.
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MODULE 3 - POLYNOMIAL FUNCTIONS SUMMATIVE
ASSIGNMENT
Complete the following questions from your text book. Show steps completely and clearly, as
marks are assigned for mathematical literacy and communication. Always use graph paper,
rulers, and pencils as necessary. Attach securely to this booklet before you hand everything in.
Module 3 is now complete. Once you have received your corrected work, review your
instructor’s comments and prepare for your module three test.
Review Page 109 Characteristics of Polynomial Functions for quick reference.
Text: Pre-Calculus 12
Section 3.1: Page 114 to 117 #4a, 4c, 4d, 4f, 6,10
Section 3.2: Page 124 to 125 #1, 3a, 3e, 4b, 4f, 5b, 6a, 8a, 11
Section 3.3: Page 133 to135 #1c, 2b, 3e, 4c, 5a, 6a, 7a, 7d, 10, 15