You are told that f(3) = 0. Which of the following is therefore a factor of f(x)?
A(x + 3)
B(x − 3)
C(3x)
D(x · f(3))
The Factor Theorem states: if c is a root (f(c) = 0), then (x − c) is a factor. With c = 3, the factor is (x − 3), not (x + 3). The sign flip is the most common error: students see the root '3' and write '+3' in the factor. Remember the factor always uses subtraction: (x minus the root).
Question 2 Multiple Choice
A student evaluates f(2) = 0 for the polynomial f(x) = x³ − 6x² + 11x − 6. What is the correct next step to find all remaining roots?
AEvaluate f at several other integers until finding values where f(c) = 0, then list all such c as roots
BDivide f(x) by (x − 2) to obtain a quotient polynomial of lower degree, then factor or solve that quotient
CApply the quadratic formula to f(x) after substituting x = 2 into the quadratic terms
DConclude that x = 2 is the only root, since the Factor Theorem was already applied
Once (x − 2) is confirmed as a factor, synthetic or long division peels it off: f(x) ÷ (x − 2) gives a degree-2 quotient that can be factored by inspection or the quadratic formula. This 'testing and peeling' process dismantles the polynomial one factor at a time. Option A is the brute-force approach and misses the efficiency of division; option D incorrectly assumes a cubic has only one real root.
Question 3 True / False
The Factor Theorem tells you which values of c to test as potential roots of a polynomial.
TTrue
FFalse
Answer: False
The Factor Theorem only confirms or refutes a candidate: if you evaluate f(c) and get 0, the theorem tells you (x − c) is a factor. It does not supply the candidates to try. That job belongs to the Rational Root Theorem, which lists all possible rational roots based on the leading coefficient and constant term. The two theorems work in tandem: Rational Root Theorem provides the candidates; Factor Theorem tests them.
Question 4 True / False
If (x − c) is a factor of f(x), then the graph of y = f(x) crosses or touches the x-axis at the point (c, 0).
TTrue
FFalse
Answer: True
This is the geometric leg of the three-way equivalence at the heart of the Factor Theorem: c is a zero of f (f(c) = 0) ↔ (x − c) is a factor of f(x) ↔ the graph of y = f(x) has an x-intercept at (c, 0). Each description says the same thing in a different language — algebraic (root), algebraic-structural (factor), and geometric (intercept).
Question 5 Short Answer
Explain the three-way equivalence stated in the Factor Theorem, connecting the concepts of a zero, a factor, and an x-intercept.
Think about your answer, then reveal below.
Model answer: For a polynomial f(x), these three statements are all equivalent — if any one is true, all three are true: (1) c is a zero of f, meaning f(c) = 0; (2) (x − c) is a factor of f(x), meaning f(x) = (x − c)·q(x) for some polynomial q with no remainder; (3) (c, 0) is an x-intercept of the graph of y = f(x). Finding a root graphically tells you a factor; confirming a factor algebraically tells you an x-intercept. The theorem bridges the algebraic and geometric representations of polynomials.
This three-way equivalence is what makes the Factor Theorem powerful rather than a mere corollary of the Remainder Theorem. It means you can move freely between representations: use a graph to spot an approximate intercept, test the nearby integer with f(c), and if it's zero, write the factor and divide it out. Each representation gives you information the others make use of.