A reliable method
Rewrite an equation as a polynomial equal to zero before using the zero-product property. Multiply candidate factors back out to verify both middle and constant terms.
The common trap
A factor is an expression, while a zero is an input. Factoring a nonzero product does not permit setting each factor equal to zero.
Make the connection
A repeated factor can touch rather than cross the horizontal axis. Record multiplicity as well as the location of a zero.
Zero versus remainder
For P(x)=x²−5x+6, P(2)=0, so x−2 is a factor; P(4)=2, so division by x−4 leaves remainder 2. The same substitution rule answers both questions, but only a zero remainder establishes a factor. The factorisation (x−2)(x−3) gives distinct zeros 2 and 3. Compare (x−2)²: its only distinct zero is 2, counted twice by multiplicity. A sum of all roots must therefore specify whether multiplicity is counted. Expanding the factors checks the coefficients, while substituting the proposed roots checks the zeros. These are complementary rather than identical checks.
Transfer, explain and check
For F(x)=(x−1)²(x+4), list zeros with multiplicity and find their sum. Check: 1,1,−4; sum −2. Explain why the sum of distinct zeros would instead be −3.
What is the remainder when P(x)=x³ + 3x + 4 is divided by x−2?
- Use the remainder theorem at x=2.
- Compute 2³+3(2)+4.
- The remainder is 18.
Compare the other answer choices
Treat the constant as every possible division remainder.
Omit the cubic contribution.
Evaluate at −2 rather than at the divisor zero 2.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus course and syllabus