SAT · Passport to Advanced Math · Polynomial expressions and factoring
Question sat-q-00082
Hard Multiple choice No calculator
The question
What is the sum of the solutions to the equation x² − 12x + 35 = 0?
- -12
- 12
- 35
- 14
Show answer & worked solution
Answer: 12
Worked solution. By Vieta's formulas, for x² + bx + c = 0 the sum of the roots is −b. Here −b = 12 = 12. (Or factor: (x − 7)(x − 5) = 0, so the roots are 7 and 5 and their sum is 12.)
Why each wrong choice is wrong. One distractor is the product of the roots (7 × 5). One is the sum with the wrong sign — a Vieta sign-flip slip. One is off by 2.
Test-day tactic. Memorize Vieta: sum of roots is −b/a, product is c/a. You can answer many SAT quadratic questions without ever solving the equation.
About this question type
Polynomial items test the relationship between the factored form and the zeros, end behavior, polynomial long division (rare), and the remainder theorem (occasionally). Most items reduce to either factoring a quadratic-shaped polynomial or recognizing that a polynomial has a factor of (x − a) when a is a zero.
You will see a question shaped like this one on roughly every other official SAT form, typically at the back end of the section, where the score gradient is steepest. Treat any miss in this subtopic as a signal to drill the subtopic page before you do another full practice test.