Advanced Math is one of the four major content areas on the Digital SAT Math section, and it covers the algebraic manipulation skills that show up constantly once equations get more complex than a simple linear expression: simplifying and combining polynomial expressions, factoring quadratics, applying exponent rules, working with rational expressions that have variables in the denominator, and solving exponential equations.
Most Advanced Math questions reward careful, methodical execution over any clever shortcut. When simplifying a polynomial expression that includes a term being subtracted and distributed, like 50x²+40x−150−20(x²+x−1), distribute the negative sign through every term inside the parentheses before combining anything, since a single sign error there cascades into a wrong answer that still "looks" reasonable. For rational expressions with unlike denominators, find a common denominator first, then combine the numerators carefully term by term. For exponent rules, remember that dividing by a variable raised to a negative power is the same as multiplying by that variable raised to the corresponding positive power, since 1/x−n = xn. And for equations with a power raised to another power, like (2m+n)m−n, multiply the exponents together first, since (ap)q = apq, before trying to match both sides to the same base.
Below are 6 SAT Math-style Advanced Math practice problems covering simplifying polynomial expressions, factoring quadratics, exponent rules, subtracting rational expressions, and solving exponential equations. Work through each problem, select your answer, and check whether you got it right. Every problem includes a complete, step-by-step explanation. Once you've worked through these, keep building your skills with the free SAT Math Question Bank, with 2,500+ more digital SAT practice problems and instant explanations.
Which answer correctly simplifies the expression
Distribute the −25 across every term inside the parentheses: −25(x²+x−2) = −25x²−25x+50.
Rewrite the full expression with this distributed: 60x²+30x−200−25x²−25x+50.
Combine like terms: (60x²−25x²) + (30x−25x) + (−200+50) = 35x²+5x−150.
The correct answer is C.
Simplify the following expression: x(x−7) + 3(x−4) − 9.
Distribute both products: x(x−7) = x²−7x, and 3(x−4) = 3x−12.
Combine everything: x²−7x+3x−12−9 = x²−4x−21.
Factor this quadratic: look for two numbers that multiply to −21 and add to −4. Those numbers are −7 and 3: (x−7)(x+3).
The correct answer is A.
Which of the following expressions is equal to
When dividing powers of the same base, subtract the exponent in the denominator from the exponent in the numerator: x−2−(−1) = x−1, and y4−3 = y1.
This gives x−1y1.
Since x−1 = 1/x, this becomes y/x.
The correct answer is A.
Keep the momentum going.
Explore 2,500+ Free SAT Math Practice ProblemsSimplify the following expression
Find a common denominator by multiplying the two denominators together: (x−4)(x+3).
Rewrite each fraction over this common denominator, then combine the numerators: [(3x−2)(x+3) − (x−2)(x−4)] / [(x−4)(x+3)].
Expand each product: (3x−2)(x+3) = 3x²+7x−6, and (x−2)(x−4) = x²−6x+8. Subtract: 3x²+7x−6−(x²−6x+8) = 2x²+13x−14.
The denominator expands to x²−x−12. The full result is (2x²+13x−14)/(x²−x−12). The correct answer is A.
The difference of −5x²+6x−8 and 4x²+14x+2 can be written in the form ax²+bx+c, where a, b, and c are constants. What is the value of a × c?
"The difference of A and B" means A − B. Subtract, distributing the negative sign across the second polynomial: (−5x²+6x−8) − (4x²+14x+2) = −5x²+6x−8−4x²−14x−2.
Combine like terms: (−5x²−4x²) + (6x−14x) + (−8−2) = −9x²−8x−10.
This matches the form ax²+bx+c with a=−9, b=−8, c=−10.
a × c = (−9)(−10) = 90. The correct answer is C.
5 down, 1 to go. Almost there.
Explore 2,500+ Free SAT Math Practice ProblemsWhat is the value of m in the equation below
if n²=12 and if m>0?
Use the power rule (ap)q = apq: the left side becomes 3(m+n)(m−n).
Since (m+n)(m−n) = m²−n² (a difference of squares), and 81 can be written as 34, this gives m²−n² = 4.
Substitute the given value n²=12: m²−12=4 → m²=16.
Since m>0, take the positive square root: m=4. The correct answer is C.
Want more practice like this?
Explore 2,500+ Free SAT Math Practice Problems