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Lesson Plans and Worksheets for Algebra II

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Common Core For Algebra

Student Outcomes

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

Worksheets for Algebra II, Module 1, Lesson 19

Classwork

Exercises 1–3

- Consider the polynomial function 𝑓(𝑥) = 3𝑥
^{2}+ 8𝑥 −4.

a. Divide 𝑓 by 𝑥 − 2. b. Find 𝑓(2). - Consider the polynomial function 𝑔(𝑥) = 𝑥
^{3}− 3𝑥^{2}+ 6𝑥 + 8.

a. Divide 𝑔 by 𝑥 + 1. b. Find 𝑔(−1). - Consider the polynomial function ℎ(𝑥) = 𝑥
^{3}+ 2𝑥 − 3.

a. Divide ℎ by 𝑥 − 3. b. Find ℎ(3)

Exercises 4–6

- Consider the polynomial 𝑃(𝑥) = 𝑥
^{3}+ 𝑘𝑥^{2}+ 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). - Consider the polynomial 𝑃(𝑥) = 𝑥
^{4}+ 3𝑥^{3}− 28𝑥^{2}− 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. - 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) = 𝑥^{2} − 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) = 𝑥^{3} − 5𝑥^{2} + 3𝑥 + 9, then 𝑃(3) = 27 − 45 + 9+ 9 = 0, so (𝑥 − 3) is a factor of 𝑃.

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