The Remainder Theorem

The Remainder Theorem

Student Outcomes

• Students know and apply the Remainder Theorem and understand the role zeros play in the theorem.

New York State Common Core Math Algebra II, Module 1, Lesson 19

Worksheets for Algebra II, Module 1, Lesson 19

Classwork

Exercises 1–3

1. Consider the polynomial function 𝑓(𝑥) = 3𝑥² + 8𝑥 −4.
   a. Divide 𝑓 by 𝑥 − 2. b. Find 𝑓(2).
2. Consider the polynomial function 𝑔(𝑥) = 𝑥³ − 3𝑥² + 6𝑥 + 8.
   a. Divide 𝑔 by 𝑥 + 1. b. Find 𝑔(−1).
3. Consider the polynomial function ℎ(𝑥) = 𝑥³ + 2𝑥 − 3.
   a. Divide ℎ by 𝑥 − 3. b. Find ℎ(3)

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Exercises 4–6

4. Consider the polynomial 𝑃(𝑥) = 𝑥³ + 𝑘𝑥² + 𝑥 + 6.
   a. Find the value of 𝑘 so that 𝑥 + 1 is a factor of 𝑃.
   b. Find the other two factors of 𝑃 for the value of 𝑘 found in part (a).
5. Consider the polynomial 𝑃(𝑥) = 𝑥⁴ + 3𝑥³ − 28𝑥² − 36𝑥 + 144.
   a. Is 1 a zero of the polynomial 𝑃?
   b. Is 𝑥 + 3 one of the factors of 𝑃?
   c. The graph of 𝑃 is shown to the right. What are the zeros of 𝑃?
   d. Write the equation of 𝑃 in factored form.
6. Consider the graph of a degree 5 polynomial shown to the right, with 𝑥-intercepts −4, −2, 1, 3, and 5.
   a. Write a formula for a possible polynomial function that the graph represents using 𝑐 as the constant factor.
   b. Suppose the 𝑦-intercept is −4. Find the value of 𝑐 so that the graph of 𝑃 has 𝑦-intercept −4.

Lesson Summary

REMAINDER THEOREM: Let 𝑃 be a polynomial function in 𝑥, and let 𝑎 be any real number. Then there exists a unique polynomial function 𝑞 such that the equation
𝑃(𝑥) = 𝑞(𝑥)(𝑥 − 𝑎)+ 𝑃(𝑎)
is true for all 𝑥. That is, when a polynomial is divided by (𝑥 − 𝑎), the remainder is the value of the polynomial evaluated at 𝑎.

FACTOR THEOREM: Let 𝑃 be a polynomial function in 𝑥, and let 𝑎 be any real number. If 𝑎 is a zero of 𝑃, then (𝑥 − 𝑎) is a factor of 𝑃.
Example: If 𝑃(𝑥) = 𝑥² − 3 and 𝑎 = 4, then 𝑃(𝑥) = (𝑥 + 4)(𝑥 − 4) + 13 where 𝑞(𝑥) = 𝑥 + 4 and 𝑃(4) = 13.
Example: If 𝑃(𝑥) = 𝑥³ − 5𝑥² + 3𝑥 + 9, then 𝑃(3) = 27 − 45 + 9+ 9 = 0, so (𝑥 − 3) is a factor of 𝑃.

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