SAT Math: Polynomial Structure and Factoring Practice Questions
Use coefficients, common factors, products, and zeros to reason about polynomial expressions. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
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1. The identity (x + a)(x + 4) = x^2 + 7x + 12 holds for every real x. What is a?
Explanation. Expanding gives x^2 + (a + 4)x + 4a. Matching x-coefficients gives a + 4 = 7, so a = 3. The constant also checks: 4a = 12.
2. When (2x - 3)(x + 5) is expanded, what is the coefficient of x?
Explanation. Multiply all four term pairs: 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15. Combining the two middle terms gives coefficient 7.
3. The expression 6x^2 + 15x is written as 3x(2x + k). What is k?
Explanation. Distributing 3x gives 6x^2 + 3kx. The x-coefficients must agree, so 3k = 15 and k = 5. This factors out the common factor 3x.
4. What is the largest real solution of x^3 - 9x = 0?
Explanation. Factor x^3 - 9x as x(x^2 - 9) = x(x - 3)(x + 3). The zeros are -3, 0, and 3. The largest is 3.
5. If x^2 - 10x + c is the square of a linear expression with leading coefficient 1, what is c?
Explanation. Write the square as (x - b)^2 = x^2 - 2bx + b^2. Matching -2b = -10 gives b = 5, so c = b^2 = 25.
6. For the polynomial p(x) = (x - 2)(x + 3)(x - 5), what is p(0)?
Explanation. Substitute x = 0 directly into the factored form: (-2)(3)(-5) = 30. This is also the constant term of the expanded polynomial.
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FAQ
How can I check a proposed factorization?
Multiply the factors and compare every coefficient with the original polynomial. Substitution can quickly disprove a match, but matching all coefficients proves the identity.
How do factors give zeros?
A product is zero when at least one factor is zero. Solve each factor equation and retain all distinct solutions.