Advanced Math · SAT Math

Polynomials on the Digital SAT

Polynomial questions on the Digital SAT cover degree, end behavior, factoring, finding zeros, and identifying polynomial identities. The math is mostly algebra you already know — the SAT just dresses it up in higher-power expressions. This lesson covers everything you need, plus 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~10 min read

Degree, leading coefficient, and end behavior

A polynomial is a sum of terms where each term is a constant times a non-negative integer power of x. Examples: 3x² - 5x + 7, x⁴ + 2x, 5 (a constant is a polynomial of degree 0).

Degree is the highest exponent. 3x⁴ - 2x² + 1 has degree 4. Degree controls how many zeros the polynomial can have (at most equal to the degree).

Leading coefficient is the number multiplying the highest-power term. In 3x⁴ - 2x² + 1, it's 3. Combined with the degree, it determines end behavior (what the graph does as x → ±∞):

Even degree, positive leading coefficient: both ends up.

Even degree, negative leading coefficient: both ends down.

Odd degree, positive leading coefficient: down-left, up-right.

Odd degree, negative leading coefficient: up-left, down-right.

Zeros, roots, and factors

The zeros (also called roots or x-intercepts) of a polynomial are the values of x where the polynomial equals 0.

The factor theorem: if x = c is a zero of polynomial p(x), then (x - c) is a factor of p(x). And vice versa: if (x - c) is a factor, then c is a zero.

Example. If p(x) = (x - 2)(x + 3)(x - 5), the zeros are x = 2, -3, 5. The graph crosses the x-axis at those three points.

Sign flip warning: in (x - 2), the zero is +2, not -2. The zero is whatever makes the factor equal to 0.

Polynomial identities to memorize

The SAT tests recognition of three structural patterns. Knowing these by sight saves real time:

Difference of squares: a² - b² = (a + b)(a - b).

Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)².

Sum / difference of cubes: a³ + b³ = (a + b)(a² - ab + b²), a³ - b³ = (a - b)(a² + ab + b²).

Example. Factor x² - 49. Recognize as difference of squares: (x + 7)(x - 7). No FOIL or trial-and-error needed.

The remainder theorem

When you divide a polynomial p(x) by (x - c), the remainder equals p(c). That's it — the entire theorem.

SAT use: instead of doing long polynomial division, just evaluate p(c) directly.

Example. What is the remainder when p(x) = x³ - 4x + 5 is divided by (x - 2)?

Plug x = 2: p(2) = 8 - 8 + 5 = 5. Remainder is 5.

Corollary: if p(c) = 0, the remainder is 0 — meaning (x - c) divides p(x) exactly, which means (x - c) is a factor (back to the factor theorem).

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Solving with Desmos

Desmos handles polynomial questions cleanly. Three techniques:

1. Find zeros by graphing

Type the polynomial as y = x³ - 6x² + 11x - 6. Desmos shows the x-intercepts as grey dots — click each one to see the exact zero. Way faster than factoring by hand.

2. Apply the remainder theorem

For "remainder when p(x) = x³ + 2x - 5 is divided by (x - 3)": just type 3³ + 2(3) - 5 into Desmos. Returns 28 — that's the remainder.

3. Confirm factoring by graphing both forms

If the question asks "which is equivalent to x² - 8x + 15?", type the original AND each answer choice. Identical graphs confirm equivalence.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Sign-flipping the zero from the factor

(x - 3) means the zero is +3, not -3. (x + 3) means the zero is -3. The zero is the value that makes the factor equal to zero — flip the sign of the constant.

Confusing 'zero' with 'y-intercept'

Zeros (or x-intercepts) are where the polynomial equals 0 — output zero. The y-intercept is the value when x = 0. Different things.

Missing a difference of squares disguised as something else

x⁴ - 16 isn't always obvious as a difference of squares. Rewrite as (x²)² - 4² = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2). Always check.

Doing polynomial long division when the remainder theorem works

If a question asks 'what is the remainder when p(x) is divided by (x - c)?', just compute p(c). Long division wastes 60+ seconds.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

What is the degree of the polynomial 5x³ - 2x⁴ + 7x - 9?

  1. 1
  2. 3
  3. 4
  4. 9

Click to reveal solution →

Answer: (C) 4

The highest exponent is 4 (on -2x⁴). Don't be thrown off by it being listed second — degree is determined by the highest power, regardless of order.

2

Which expression is equivalent to x² - 64?

  1. (x - 8)²
  2. (x + 8)(x - 8)
  3. (x - 16)(x + 4)
  4. (x + 16)(x - 4)

Click to reveal solution →

Answer: (B) (x + 8)(x − 8)

Difference of squares: a² - b² = (a + b)(a - b). Here a = x, b = 8.

3

If p(x) = (x - 2)(x + 3)(x - 5), what are the zeros of p?

  1. -2, 3, -5
  2. 2, -3, 5
  3. -2, -3, -5
  4. 2, 3, 5

Click to reveal solution →

Answer: (B) 2, −3, 5

Each factor (x - c) gives a zero at x = c. Note the sign flips: (x + 3) gives x = -3.

4

What is the remainder when p(x) = x³ + 2x - 5 is divided by (x - 1)?

  1. -5
  2. -2
  3. 0
  4. -2

Click to reveal solution →

Answer: (B) −2

By the remainder theorem, the remainder is p(1) = 1 + 2 - 5 = -2.

5

Which expression is equivalent to (x + 5)² - 25?

  1. x²
  2. x² + 10x
  3. x² + 10x + 50
  4. x² - 10x

Click to reveal solution →

Answer: (B) x² + 10x

Expand (x + 5)²: x² + 10x + 25.

Subtract 25: x² + 10x.

6

A polynomial p(x) has zeros at x = -3 and x = 4. Which is a factor of p(x)?

  1. (x - 3)
  2. (x + 4)
  3. (x + 3)(x - 4)
  4. (x - 3)(x + 4)

Click to reveal solution →

Answer: (C) (x + 3)(x − 4)

Zero at x = -3 means factor (x + 3). Zero at x = 4 means factor (x - 4). Their product divides p(x).

7

What is the y-intercept of p(x) = 2x³ - 5x² + 3x - 8?

  1. -8
  2. -5
  3. 2
  4. 3

Click to reveal solution →

Answer: (A) −8

The y-intercept is p(0): every term with x vanishes, leaving the constant -8.

8

If (x - 2) is a factor of p(x) = x³ + ax² - 12x + 4, what is the value of a?

Click to reveal solution →

Answer: a = 3

If (x - 2) is a factor, then p(2) = 0.

p(2) = 2³ + a(2²) - 12(2) + 4 = 8 + 4a - 24 + 4 = 4a - 12.

Set to 0: 4a - 12 = 0 → a = 3.

Frequently asked questions

What is a polynomial?

+
A polynomial is an expression made of terms, each of which is a constant times a non-negative integer power of a variable. Examples include 3x² - 5x + 7 and x⁴ - 1. The degree is the highest exponent.

What is the factor theorem?

+
If c is a zero of polynomial p(x), then (x − c) is a factor of p(x). And conversely: if (x − c) is a factor, then c is a zero. This lets you go between zeros and factors of a polynomial.

What is the remainder theorem?

+
When you divide a polynomial p(x) by (x − c), the remainder equals p(c). So instead of doing polynomial long division, just plug c into the polynomial.

What's the difference between zeros, roots, and x-intercepts?

+
They all mean the same thing — the values of x where the polynomial equals zero. The graph crosses (or touches) the x-axis at each one.

How do I factor a difference of squares?

+
Use a² − b² = (a + b)(a − b). For example, x² − 49 = (x + 7)(x − 7) and x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2).

Keep going

Quadratics

Polynomials of degree 2 — a special case

All SAT Math topics

Browse every Digital SAT math topic

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