The Special Role of Zero in Factoring

The Special Role of Zero in Factoring

Student Outcomes

• Students find solutions to polynomial equations where the polynomial expression is not factored into linear factors.
• Students construct a polynomial function that has a specified set of zeros with stated multiplicity.

New York State Common Core Math Algebra II, Module 1, Lesson 11

Worksheets for Algebra 2

Classwork

Opening Exercise

Find all solutions to the equation (𝑥² + 5𝑥 + 6)(𝑥² − 3𝑥 − 4) = 0

Exercise 1

1. Find the solutions of (𝑥² − 9)(𝑥² −16) = 0.

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Example 1:

Suppose we know that the polynomial equation 4𝑥³ − 12𝑥² +3𝑥 + 5 = 0 has three real solutions and that one of the factors of 4𝑥³ − 12𝑥² +3𝑥 + 5 is (𝑥 − 1). How can we find all three solutions to the given equation?

Exercises 2–5

2. Find the zeros of the following polynomial functions, with their multiplicities.
   a. 𝑓(𝑥) = (𝑥 + 1)(𝑥 − 1)(𝑥² + 1)
   b. 𝑔(𝑥) = (𝑥 − 4)³(𝑥 − 2)⁸
   c. ℎ(𝑥) = (2𝑥 − 3)⁵
   d. 𝑘(𝑥) = (3𝑥 + 4)¹⁰⁰(𝑥 − 17)⁴
3. Find a polynomial function that has the following zeros and multiplicities. What is the degree of your polynomial?
4. Is there more than one polynomial function that has the same zeros and multiplicities as the one you found in Exercise 3?
5. Can you find a rule that relates the multiplicities of the zeros to the degree of the polynomial function?

Relevant Vocabulary Terms

In the definitions below, the symbol ℝ stands for the set of real numbers.

FUNCTION: A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is assigned to one and only one element of 𝑌. The set 𝑋 in the definition above is called the domain of the function. The range (or image) of the function is the subset of 𝑌, denoted 𝑓(𝑋), that is defined by the following property: 𝑦 is an element of 𝑓(𝑋) if and only if there is an 𝑥 in 𝑋 such that 𝑓(𝑥) = 𝑦. If 𝑓(𝑥) = 𝑥² where 𝑥 can be any real number, then the domain is all real numbers (denoted ℝ), and the range is the set of nonnegative real numbers.
POLYNOMIAL FUNCTION: Given a polynomial expression in one variable, a polynomial function in one variable is a function 𝑓: ℝ → ℝ such that for each real number 𝑥 in the domain, 𝑓(𝑥) is the value found by substituting the number 𝑥 into all instances of the variable symbol in the polynomial expression and evaluating. It can be shown that if a function 𝑓: ℝ → ℝ is a polynomial function, then there is some non-negative integer 𝑛 and collection of real numbers 𝑎₀, 𝑎₁, 𝑎₂,… , 𝑎_𝑛 with 𝑎_𝑛 ≠ 0 such that the function satisfies the equation 𝑓(𝑥) = 𝑎_𝑛𝑥^𝑛 + 𝑎_𝑛−1𝑥^𝑛−1 + ⋯ + 𝑎₁𝑥 + 𝑎₀, for every real number 𝑥 in the domain, which is called the standard form of the polynomial function. The function 𝑓(𝑥) = 3𝑥³ +4𝑥² + 4𝑥 +7, where 𝑥 can be any real number, is an example of a function written in standard form.
DEGREE OF A POLYNOMIAL FUNCTION: The degree of a polynomial function is the degree of the polynomial expression used to define the polynomial function. The degree of 𝑓(𝑥) = 8𝑥³ + 4𝑥² + 7𝑥 +6 is 3, but the degree of 𝑔(𝑥) = (𝑥 + 1)2 − (𝑥 −1)2 is 1 because when 𝑔 is put into standard form, it is 𝑔(𝑥) = 4𝑥.
CONSTANT FUNCTION: A constant function is a polynomial function of degree 0. A constant function is of the form 𝑓(𝑥) = 𝑐, for a constant 𝑐.
LINEAR FUNCTION: A linear function is a polynomial function of degree 1. A linear function is of the form 𝑓(𝑥) = 𝑎𝑥 + 𝑏, for constants 𝑎 and 𝑏 with 𝑎 ≠ 0
QUADRATIC FUNCTION: A quadratic function is a polynomial function of degree 2. A quadratic function is in standard form if it is written in the form 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, for constants 𝑎, 𝑏, 𝑐 with 𝑎 ≠ 0 and any real number 𝑥.
CUBIC FUNCTION: A cubic function is a polynomial function of degree 3. A cubic function is of the form 𝑓(𝑥) = 𝑎𝑥3 + 𝑏𝑥 2 + 𝑐𝑥 +𝑑, for constants 𝑎, 𝑏, 𝑐,𝑑 with 𝑎 ≠ 0.
ZEROS OR ROOTS OF A FUNCTION: A zero (or root) of a function 𝑓: ℝ → ℝ is a number 𝑥 of the domain such that 𝑓(𝑥) = 0. A zero of a function is an element in the solution set of the equation 𝑓(𝑥) = 0.

Lesson Summary

Given any two polynomial functions 𝑝 and 𝑞, the solution set of the equation 𝑝(𝑥)𝑞(𝑥) = 0 can be quickly found by solving the two equations 𝑝(𝑥) = 0 and 𝑞(𝑥) = 0 and combining the solutions into one set. The number 𝑎 is a zero of a polynomial function 𝑝 with multiplicity 𝑚 if the factored form of 𝑝 contains (𝑥 − 𝑎) �

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