KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 02 · Lesson 02

Zeros, Factors & Multiplicity

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Zeros, Factors & Multiplicity

U02 · L02
1

State the end behavior of f(x)=−3x⁴+2x.

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
Answer: Both ends decrease toward −∞
Reasoning: Even degree with negative leading coefficient.
2

Find the zeros of (x−2)²(x+5).

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
Answer: x=2 (multiplicity 2), x=−5
Reasoning: Set each factor equal to zero.
3

How does the graph behave at x=2 for (x−2)²(x+5)?

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
Answer: It touches and turns
Reasoning: Even multiplicity produces a touch.
4

Divide x³−4x²+x+6 by x−2.

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
Answer: x²−2x−3
Reasoning: Synthetic division with 2 gives coefficients 1,−2,−3 and remainder 0.
5

Find the remainder when f(x)=x³+2x−1 is divided by x−3.

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
Answer: 32
Reasoning: Remainder theorem: f(3)=27+6−1=32.
6

What is the maximum possible number of turning points of a degree-6 polynomial?

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
Answer: 5
Reasoning: A degree n polynomial has at most n−1 turning points.
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