Polynomial functions extend everything you know about quadratics to higher degrees, and the SAT tests them through end behavior, zeros, and factoring. This guide explains how degree and leading coefficient control a graph's shape, then walks through two complete worked examples using two different factoring strategies.
What Makes a Function a Polynomial
A polynomial function is any function built from terms like \(ax^n\) added together, where \(n\) is a non-negative whole number: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0\). Quadratics are degree-2 polynomials; the SAT also tests degree-3 (cubic) and occasionally degree-4 functions under the nonlinear functions subtopic. The higher the degree, the more times the graph can change direction, and the more zeros it can potentially have.
Degree and End Behavior
"End behavior" describes what happens to \(f(x)\) as \(x\) moves far to the left or right on the graph. Two facts control it completely: the degree (odd or even) and the sign of the leading coefficient.
| Degree | Leading coefficient | Left end | Right end |
|---|---|---|---|
| Even | Positive | Up | Up |
| Even | Negative | Down | Down |
| Odd | Positive | Down | Up |
| Odd | Negative | Up | Down |
A degree-3 polynomial with a positive leading coefficient, for example, always falls on the far left and rises on the far right — you can answer an end-behavior question without graphing anything, just by checking degree and sign.
Zeros and Multiplicity
The zeros of a polynomial are the \(x\)-values where \(f(x)=0\), found by factoring completely. A zero's multiplicity — how many times its factor repeats — changes how the graph behaves there:
- A zero with odd multiplicity (like 1 or 3) crosses straight through the \(x\)-axis.
- A zero with even multiplicity (like 2) touches the \(x\)-axis and bounces back without crossing.
- The total number of zeros, counted with multiplicity, never exceeds the degree of the polynomial.
Worked Example: Factoring Out a GCF First
Find all zeros of \(f(x) = x^3 - 4x^2 + 4x\).
- Factor out the greatest common factor first: \(f(x) = x(x^2 - 4x + 4)\).
- Factor the remaining trinomial: \(x^2-4x+4\) is a perfect square, \((x-2)^2\).
- Write the fully factored form: \(f(x) = x(x-2)^2\).
- Set each factor equal to zero: \(x = 0\) or \((x-2)^2 = 0\).
- Solve: \(x=0\) or \(x=2\).
- Since \((x-2)^2\) has multiplicity 2, the graph touches the \(x\)-axis at \(x=2\) rather than crossing it; at \(x=0\) it crosses normally.
Always pull out a common factor first — it's the single most time-saving step in polynomial factoring and the one students skip most often.
Worked Example: Factoring by Grouping
Find all zeros of \(f(x) = x^3 + 2x^2 - 9x - 18\).
- Check for a GCF across all four terms: there isn't one, since the coefficients share no common factor. This signals grouping instead.
- Split the polynomial into two pairs: \((x^3 + 2x^2) + (-9x - 18)\).
- Factor each pair separately: \(x^2(x+2) - 9(x+2)\).
- Both terms now share the factor \((x+2)\), so factor it out: \((x+2)(x^2-9)\).
- Recognize \(x^2-9\) as a difference of squares: \((x+2)(x-3)(x+3)\).
- Set each factor to zero: \(x=-2\), \(x=3\), or \(x=-3\) — three distinct zeros, each with multiplicity 1, so the graph crosses the \(x\)-axis at all three points.
Grouping is the go-to strategy whenever a cubic has four terms and no shared GCF — split into pairs, factor each pair, then look for a shared binomial factor between them.
Graphing Polynomials and Checking With Desmos
Once you've factored a polynomial and found its zeros, the embedded Desmos calculator is an excellent way to confirm you got the multiplicity right. Graph the function, zoom in on each zero, and check whether the curve crosses straight through (odd multiplicity) or bounces back without crossing (even multiplicity). This is also the fastest way to double-check end behavior — rather than reasoning through degree and leading coefficient by hand, you can simply look at which way the graph points on both sides of the screen. On a grid-in question that asks for a single zero rather than all of them, graphing first to see roughly where the zeros sit can help you catch an arithmetic mistake before you commit to an answer.
Connecting Polynomials to Equivalent Expressions
Every polynomial factoring problem is really an application of the identities from equivalent expressions — difference of squares, perfect square trinomials, and grouping. If factoring a cubic feels unfamiliar, it usually means the underlying quadratic identities need review first, not that cubic factoring is a separate skill to learn from scratch.
Polynomial questions also frequently overlap with function notation and transformations — a question might give you \(f(x)\) and ask for \(f(x-2)\) or \(2f(x)\), which shifts or stretches the same graph you just analyzed.
How to Practice This Efficiently
Because polynomial questions combine several sub-skills — factoring, end behavior, multiplicity, and graph reading — targeted, timed repetition works better than general review. Our SAT Math practice sets cover Nonlinear Functions at Easy, Medium, and Hard levels with about 22 questions each and full explanations, so you can isolate exactly which sub-skill needs more work.
Frequently asked questions
Do I need to know synthetic division for the SAT?
Rarely necessary — most SAT polynomial questions are structured so that factoring by GCF, grouping, or recognizing an identity gets you to the answer without long or synthetic division.
How many zeros can a cubic function have?
Up to three real zeros, counting multiplicity, though it can also have just one real zero with the other two being complex (non-real).
What's the fastest way to find end behavior?
Check the degree (odd or even) and the sign of the leading coefficient, then use the end-behavior table — no graphing or calculation required.
How do I decide between factoring by GCF and factoring by grouping?
Check for a common factor across every term first. If one exists, pull it out. If there isn't one but the polynomial has four terms, try grouping them into two pairs and factoring each pair separately before looking for a shared binomial.