SAT polynomials: read zeros, factors, and turning points from a graph
Every x-intercept of a polynomial is a factor in disguise. Learn to move between factors, zeros, and graphs, spot where a graph bounces instead of crossing, and use the remainder theorem shortcut.
SPSAT Practice Editorial8 min read
The short version
What you'll take away
- If p(a) = 0, then (x − a) is a factor of p(x), and the graph crosses or touches the x-axis at x = a.
- A factor that appears an even number of times makes the graph touch the axis and turn back; an odd number makes it cross.
- The remainder when p(x) is divided by x − a equals p(a).
In this article
Factors, zeros, and x-intercepts are the same information
Consider \(\displaystyle p(x) = (x + 2)(x - 1)(x - 3)\). A product is zero when one of its factors is zero, so \(\displaystyle p(x) = 0\) when \(\displaystyle x = -2\), \(\displaystyle x = 1\), or \(\displaystyle x = 3\). Those zeros are exactly where the graph meets the x-axis.
The y-intercept comes from substituting \(\displaystyle x = 0\): \(\displaystyle p(0) = (2)(-1)(-3) = 6\).
The study lab
y = (x + 2)(x − 1)(x − 3)
A cubic polynomial with three zeros and a y-intercept of 6.
- y = (x + 2)(x − 1)(x − 3)
Touch or cross: multiplicity
If a factor repeats, its zero has a higher multiplicity. In \(\displaystyle y = (x + 2)(x - 1)^2\), the factor \(\displaystyle x - 1\) appears twice. At \(\displaystyle x = 1\), the graph touches the axis and turns back instead of crossing it. At \(\displaystyle x = -2\), where the factor appears once, it crosses.
The study lab
y = (x + 2)(x − 1)²
A single zero at −2 (crosses) and a double zero at 1 (touches).
- y = (x + 2)(x − 1)²
The remainder theorem shortcut
When a polynomial \(\displaystyle p(x)\) is divided by \(\displaystyle x - a\), the remainder is \(\displaystyle p(a)\). So “p(x) divided by x − 2 leaves a remainder of 7” means \(\displaystyle p(2) = 7\). A remainder of 0 means \(\displaystyle x - a\) is a factor.
The SAT often asks which expression must be a factor given a value, or which value is a zero given a factor. Translate between them with the sign flip: zero at \(\displaystyle -4\) means a factor of \(\displaystyle x + 4\).
Your turn
Quick practice
For a polynomial p, p(−4) = 0. Which of the following must be a factor of p(x)?
Hint: Write the factor as x − a with a = −4.
Practice: try these before checking the answers
Factor first whenever you can.
- 1. What are the zeros of \(\displaystyle p(x) = x(x - 5)(x + 1)\)?
- 2. Factor \(\displaystyle x^3 - 4x\) and find its zeros.
- 3. If \(\displaystyle p(3) = 0\), name a factor of \(\displaystyle p(x)\).
- 4. When p(x) is divided by x + 1, the remainder is 4. What is p(−1)?
- 5. Where does \(\displaystyle y = (x - 2)^2(x + 1)\) touch the x-axis, and where does it cross?
Sources & further reading
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