# 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)

Exercises 4–6

1. 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).
2. 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.
3. 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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