Passport to Advanced Math - Sample SAT Math Questions - Calculator & No Calculator

Sep 14, 2026

Uploaded by Quinn

Sample SAT Math Questions: Passport to Advanced Math provides step-by-step instructional solutions for multiple exam-style problems. It includes tasks involving polynomial functions and zeros, using factorization and constant-term relationships to solve for a parameter, alongside incorrect-choice analysis that highlights common misconceptions. It also covers trigonometric problem-solving using triangle properties and radian measure, including converting between sine and cosine. A rational equation problem demonstrates clearing denominators via a common denominator, factoring the resulting quadratic, and checking for extraneous solutions.

LanguageEnglish

CategoryExam

FormatPDF

Pages3

Reading Time8 min

AccessFree to download

Sample SAT Math Questions

Passport to Advanced Math

Additional Topics in Math

Student-Produced Response Problem

SAMPLE SAT MATH QUESTIONS Passport to Advanced Math Calculator 1. The function fis defined by f (x) = 2x3 + 3x2 + cx + 8, where c is a constant. In the xy-plane, the graph off intersects the x-axis at the three points (−4, 0), ( ~~ ~~12 ,0), and (p, 0 ). What is the value of c? A) –18 B) –2 C) 2 D) 10 Students could tackle this problem in many different ways, but the focus is on their understanding of the zeros of a polynomial function and how they are used to construct algebraic representations of polynomials. Choice A is correct. The given zeros can be used to set up an equation to solve for c. Substituting –4 for x and 0 for y yields –4c = 72, or c = –18. Alternatively, since –4, 12 , and p are zeros of the polynomial function f(x) = 2x3 + 3x2 + cx + 8, it follows that f(x) = (2x − 1)(x + 4)(x − p) . Were this polynomial multiplied out, the constant term would be (−1)(4)(− p) = 4 p. (We can see this without performing the full expansion.) Since it is given that this value is 8, it goes that 4p = 8 or rather, p = 2. Substituting 2 for p in the polynomial function yields f(x) = (2x − 1)(x + 4)(x − 2), and after multiplying the factors one finds that the coefficient of the x term, or the value of c, is –18. Choice B is not the correct answer. This value is a misunderstood version of the value of p , not c, and the relationship between the zero and the factor (if a is the zero of a polynomial, its corresponding factor is x – a) has been confused. Choice C is not the correct answer. This is the value of p , not c. Using this value as the third factor of the polynomial will reveal that the value of c is –18. Choice D is not the correct answer. This represents a sign error in the final step in determining the value of c. THE RE DESIGNED SAT » FACILITATOR’S GUIDE HANDOUT 46 SAMPLE SAT MATH QUESTIONS 2. Additional Topics in Math Calculator Student-Produced Response Problem Note: Figure not drawn to scale. What is the value of cos x? This problem requires students to make use of properties of triangles to solve a problem. Because the triangle is isosceles, constructing a perpendicular from the top vertex to the opposite side will bisect the base and create two smaller right triangles. In a right triangle, the cosine of an acute angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. This 2 simplified to cos x = 3. Passport to Advanced Math No calculator gives cos x = 16~~24 ~~ , which can be 3. What is one possible solution to the equation ~~ ~~x2~~ ~~1-x1-21 = 1? Students should look for the best solution methods for solving rational equations before they begin. Looking for structure and common denominators will prove very useful at the onset and will help prevent complex computations that do not lead to a solution. In this problem, multiplying both sides of the equation by the common denominator (x + 1)(x – 1) yields 24(x – 1)– 12(x + 1) = (x + 1)(x – 1) . Multiplication and simplification then yields 12x – 36 = x2 – 1, or x2 – 12x + 35 = 0. Factoring the quadratic gives (x – 5)(x – 7) = 0, so the solutions occur at x = 5 and x = 7, both of which should be checked in the original equation to ensure that they are not extraneous. In this case, both values are solutions. THE RE DESIGNED SAT » FACILITATOR’S GUIDE HANDOUT 47 SAMPLE SAT MATH QUESTIONS Additional Topics in Math No Calculator 4. Which of the following is equal to sin A) -cos( ~~ ~~π5~~ ~~ ) B) -sin ( ~~ ~~π5~~ ~~ ) C) cos ( ~~ ~~31π0~~ ~~ ) D) sin ( ~~ ~~71π0~~ ~~ ) ( ~~ ~~π5~~ ~~ ) ? This question is solved most efficiently when a student is fluent with radian measure and has a conceptual understanding of the relationship between the sine and cosine functions. Choice C is correct. Sine and cosine are related by the equation: sin(x) = cos ( ~~ ~~ − x) . Therefore, sin ( ~~ ~~~~ ~~ ) = cos ( ~~ ~~ − ~~ ~~ ), which reduces to cos ( ~

FAQs

How do you find the constant c in the polynomial function problem?

Use the given x-intercepts (zeros) to set up an equation for c, or express the polynomial as a factored form from its zeros and match the constant term.

What is the key idea for solving the cos(x) triangle-based problem?

Recognize the triangle as isosceles, construct a perpendicular to split it into right triangles, then use the right-triangle cosine definition with adjacent and hypotenuse lengths.

How are solutions found for the rational equation problem and checked?

Multiply both sides by the common denominator to clear fractions, simplify to a quadratic, factor to get candidate solutions, then substitute back into the original equation to confirm they are not extraneous.

Sep 14, 2026

Uploaded by Quinn