Polynomial questions on Digital SAT Math cluster around a few core ideas: factoring quadratics and higher-degree expressions, using x-intercepts and function notation together (if f has an x-intercept at a, then f(a) = 0), finding unknown constants from points a graph passes through, and reasoning about the sign of a factored polynomial across different intervals. The algebra involved is rarely advanced — what separates a fast solve from a slow one is recognizing structural shortcuts instead of expanding everything out by brute force.
The 21 problems below cover this full range, including several "evaluate the function using symmetry in the expression" problems that look intimidating but collapse quickly once you notice a repeated squared term, plus x-intercept reasoning, factoring, and one exponential-growth table question. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. A few of these problems (marked in the explanations) have more than one correct numerical answer, since the underlying equation has multiple positive solutions — on the actual Digital SAT, entering any one correct value for this type of grid-in question is accepted. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
Which expression is equivalent to x² − 5x − 24?
Find two numbers that multiply to −24 and add to −5. Testing factor pairs of 24: 3 and −8 work, since 3 × (−8) = −24 and 3 + (−8) = −5.
That gives (x + 3)(x − 8). The correct answer is C.
The table shows the exponential relationship between the time, t, in years, since Sam opened a savings account and the total amount, n, in dollars, in the account: (0, 604.00), (1, 607.62), (2, 611.27). If Sam made no additional deposits or withdrawals, which equation best represents the relationship between t and n?
The starting amount at t = 0 is $604.00, so that must be the leading coefficient of the exponential model.
Check the growth rate: 607.62/604.00 ≈ 1.006, meaning the account grows by about 0.6% per year. That gives n = 604(1 + 0.006)^t. The correct answer is B.
The function f is defined by f(x) = (x − 5)(x − 8)(x + k), where k is a constant. In the xy-plane, the graph of y = f(x) passes through the point (−2, 0). What is the value of f(0)?
Since the graph passes through (−2, 0), f(−2) = 0. Neither (x − 5) nor (x − 8) equals zero at x = −2, so the factor (x + k) must be the one that's zero: −2 + k = 0, giving k = 2.
So f(x) = (x − 5)(x − 8)(x + 2). Evaluate at x = 0: f(0) = (−5)(−8)(2) = 80. The correct answer is D.
7x²(4x − 13)(4x − u) = 0. In the given equation, u is a positive constant. The sum of the solutions to the equation is 9. What is the value of u?
The distinct solutions are x = 0 (from the 7x² factor), x = 13/4 (from 4x − 13 = 0), and x = u/4 (from 4x − u = 0).
Sum these distinct solutions: 0 + 13/4 + u/4 = 9. Multiply everything by 4: 13 + u = 36, so u = 23.
The graph of the polynomial function f in the xy-plane, where y = f(x), has x-intercepts of (−7, 0) and (9, 0). Which of the following must be true?
An x-intercept at (−7, 0) means that when x = −7, the output of the function is 0 — by definition, that's exactly what f(−7) = 0 states.
The other options either use the wrong input or the wrong output. The correct answer is C.
(4/7)(4x + 7)(x + √(4k + 7))(x − √(4k + 7)) = 0. In the given equation, k is a positive constant. The product of the solutions to the equation is 77. What is the value of k?
The three solutions are x = −7/4, x = √(4k+7), and x = −√(4k+7). Multiply all three: (−7/4) × √(4k+7) × (−√(4k+7)) = (−7/4) × (−(4k+7)) = (7/4)(4k+7).
Set this equal to 77: (7/4)(4k+7) = 77, so 4k+7 = 44, giving 4k = 37 and k = 37/4.
Which of the following expressions has a factor of x + 2b, where b is a positive integer constant?
Write 3x² + Cx + 18b as 3(x + 2b)(x + q). Expanding gives 3x² + 3(2b + q)x + 6bq, so matching constant terms: 6bq = 18b, meaning q = 3 (regardless of b). Matching the x-coefficient: C = 3(2b + 3) = 6b + 9.
Testing each option's C value for a positive integer b: only C = 39 gives a whole number, since 6b + 9 = 39 leads to b = 5. The correct answer is D.
The graph of the function f in the xy-plane, where y = f(x), has x-intercepts of (−5, 0), (3, 0), and (6, 0). Which of the following must be true?
An x-intercept at (6, 0) means f(6) = 0 by definition. Note that 5 is not one of the listed intercepts (the intercepts are −5, 3, and 6), so f(5) = 0 is not guaranteed.
The correct answer is D.
34z¹⁴ + bz⁵ + 70. In the given expression, b is a positive integer. If qz⁵ + r is a factor of the expression, where q and r are positive integers, what is the greatest possible value of b?
Treat this as a quadratic in z⁵: 34Z² + bZ + 70, which factors as (qZ + r)(sZ + t) with qs = 34 and rt = 70, giving b = qt + rs.
To maximize b, pair the largest factors together: use q = 1, s = 34 (from 34 = 1 × 34) and r = 70, t = 1 (from 70 = 70 × 1). This gives b = (1)(1) + (70)(34) = 1 + 2,380 = 2381. Checking: (z⁵ + 70)(34z⁵ + 1) = 34z¹⁴ + 2381z⁵ + 70. ✓
Which expression is a factor of 2x² + 30x + 8?
Look for a common factor shared by all three terms: 2x², 30x, and 8.
The greatest common numeric factor of 2, 30, and 8 is 2, and it divides every term evenly. None of the x-containing options (4x, 30x, 2x²) can divide evenly into the constant term 8. The correct answer is A.
The function p is defined by p(x) = ((x + 6)² − b)((x + 6)² − c), where a, b, and c are constants. In the xy-plane, the graph of y = p(x) passes through the points (−7, 24) and (0, 899). What is the value of p(−12) + p(−5)?
Notice that (x + 6)² only depends on the distance from x to −6. Since −12 is 6 away from −6, and 0 is also 6 away from −6, (−12+6)² = (0+6)² = 36 — meaning p(−12) = p(0) = 899.
Similarly, −5 is 1 away from −6, and −7 is also 1 away from −6, so (−5+6)² = (−7+6)² = 1 — meaning p(−5) = p(−7) = 24. So p(−12) + p(−5) = 899 + 24 = 923.
The function f is defined by f(x) = 3(x − a)(x − b)(x − c), where a, b, and c are distinct constants. When a < x < b, the value of f(x) is positive. The graph of y = f(x) in the xy-plane contains the point (r, s), where r and s are constants. If s = 8, which of the following could be true? I. r < a II. b < r < c III. r > c
Since f is positive specifically on the interval (a, b), the roots must be ordered a < b < c (a cubic with positive leading coefficient is negative before the smallest root, positive between the smallest and middle root, negative between the middle and largest root, and positive after the largest root).
With that ordering, f(x) is negative for r < a (ruling out I) and negative for b < r < c (ruling out II), but positive for r > c — consistent with s = 8. The correct answer is B.
The graph of a polynomial function y = f(x) in the xy-plane passes through the point (8, 9). What is the value of f(8)?
A graphed point (8, 9) directly means that when x = 8, the function's output is 9.
That's exactly what f(8) represents. The correct answer is B.
The function f is defined by f(x) = (x − 8)(x − 5)(x − 3). In the xy-plane, the graph of y = g(x) is the result of translating the graph of y = f(x) up 3 units. What is the value of g(0)?
Translating a graph up 3 units means g(x) = f(x) + 3 for every x.
First find f(0) = (−8)(−5)(−3) = −120. Then g(0) = f(0) + 3 = −120 + 3 = −117.
0 = (x − k)(7x + t) − (x − k). In the given equation, k and t are positive constants. A solution to the equation is x = −22/7. What is the value of t?
Factor out the common (x − k) term: 0 = (x − k)[(7x + t) − 1] = (x − k)(7x + t − 1). This gives two possible solutions: x = k, or x = (1 − t)/7.
Since k is stated to be positive but the given solution −22/7 is negative, it must come from the second factor: (1 − t)/7 = −22/7, so 1 − t = −22, giving t = 23.
The function h is defined by h(x) = (x + p)(x − 2)(2x − 12), where p is a constant. In the xy-plane, the graph of y = h(x) passes through the point (−3, 0). What is the value of h(0)?
Since h(−3) = 0, and neither (x − 2) nor (2x − 12) equals zero at x = −3, the factor (x + p) must be zero there: −3 + p = 0, so p = 3.
So h(x) = (x + 3)(x − 2)(2x − 12). Evaluate at x = 0: h(0) = (3)(−2)(−12) = 72. The correct answer is D.
The graph of the polynomial function f in the xy-plane, where y = f(x), has x-intercepts of (−5, 0) and (6, 0). Which of the following must be true?
An x-intercept at (−5, 0) means f(−5) = 0 by definition — that's what an x-intercept is.
The correct answer is C.
(x − 16)(x − 10)(x + 7)(x + 17) = 0. What is the positive solution to the given equation?
Set each factor equal to zero: x = 16, x = 10, x = −7, or x = −17.
Two of these solutions are positive: x = 16 and x = 10 (on the actual Digital SAT, a grid-in question like this accepts either correct positive value).
(x − 16)(x − 12)(x + 7)(x + 13) = 0. What is the positive solution to the given equation?
Set each factor equal to zero: x = 16, x = 12, x = −7, or x = −13.
Two of these solutions are positive: x = 16 and x = 12 (on the actual Digital SAT, a grid-in question like this accepts either correct positive value).
The graph of the polynomial function f in the xy-plane, where y = f(x), passes through the point (5, 3). What is the value of f(5)?
A graphed point (5, 3) directly means that when x = 5, the function's output is 3.
That's exactly what f(5) represents. The correct answer is A.
The function p is defined by p(x) = a((x + 5)² − b)((x + 5)² − c), where a, b, and c are constants. In the xy-plane, the graph of y = p(x) passes through the points (−6, 30) and (0, 342). What is the value of p(−10) + p(−4)?
Since (x + 5)² depends only on the distance from x to −5: −10 is 5 away from −5, and 0 is also 5 away from −5, so (−10+5)² = (0+5)² = 25 — meaning p(−10) = p(0) = 342.
Similarly, −4 is 1 away from −5, and −6 is also 1 away from −5, so (−4+5)² = (−6+5)² = 1 — meaning p(−4) = p(−6) = 30. So p(−10) + p(−4) = 342 + 30 = 372.
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