State the end behavior of f(x)=−3x⁴+2x.
Answer: Both ends decrease toward −∞
Reasoning: Even degree with negative leading coefficient.
Reasoning: Even degree with negative leading coefficient.
Teacher Key · Unit 02 · Lesson 04
Polynomial Division & the Remainder Theorem
Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.
Khaemenes Academy · Teacher Key
Polynomial Division & the Remainder Theorem
Find the zeros of (x−2)²(x+5).
Answer: x=2 (multiplicity 2), x=−5
Reasoning: Set each factor equal to zero.
Reasoning: Set each factor equal to zero.
How does the graph behave at x=2 for (x−2)²(x+5)?
Answer: It touches and turns
Reasoning: Even multiplicity produces a touch.
Reasoning: Even multiplicity produces a touch.
Divide x³−4x²+x+6 by x−2.
Answer: x²−2x−3
Reasoning: Synthetic division with 2 gives coefficients 1,−2,−3 and remainder 0.
Reasoning: Synthetic division with 2 gives coefficients 1,−2,−3 and remainder 0.
Find the remainder when f(x)=x³+2x−1 is divided by x−3.
Answer: 32
Reasoning: Remainder theorem: f(3)=27+6−1=32.
Reasoning: Remainder theorem: f(3)=27+6−1=32.
What is the maximum possible number of turning points of a degree-6 polynomial?
Answer: 5
Reasoning: A degree n polynomial has at most n−1 turning points.
Reasoning: A degree n polynomial has at most n−1 turning points.