Math, Advanced Math, Stage 3
Polynomials on the SAT: lesson and practice
To subtract polynomials, change the sign of every term in the second bracket. To multiply, multiply every term by every term. The zeros of a polynomial come from its factors: (x − 3) gives the zero x = 3. The remainder when p(x) is divided by (x − a) is p(a).
Loading practice…
| Task | Example |
|---|---|
| Subtract | (3x² + 2x) − (x² − 5x) = 2x² + 7x |
| Multiply | (x + 3)(x − 5) = x² − 2x − 15 |
| Zeros | p(x) = 2(x − 4)(x + 1) → 4 and −1 |
| Remainder | p(x) ÷ (x − 2) leaves p(2) |
| Simplify | (x² − 9)/(x − 3) = x + 3 |
Check any "equivalent expression" answer by substituting x = 2 into both expressions.
Common mistakes
Not distributing the minus sign−(x² − 5x) = −x² + 5x.
Missing the middle term(x + 3)(x − 5) has an x-term: −5x + 3x.
Cancelling terms, not factorsFactor first, then cancel whole factors only.
Sample questions with answers
1. Which is equivalent to (3x² − 5x) − (2x² − x)?
- x² − 6x
- x² − 4x (correct)
- 5x² − 6x
- x² + 4x
−(2x² − x) = −2x² + x, so the result is x² − 4x.
2. Which is a zero of p(x) = 2(x − 2)(x + 1)²?
- 1
- −1 (correct)
- 2x
- −2
x + 1 = 0 when x = −1 (2 is also a zero).
3. Which is equivalent to (x² − 6x + 8)/(x − 4)?
- x − 6
- x + 2
- x − 2 (correct)
- x − 4
x² − 6x + 8 = (x − 4)(x − 2).
Frequently asked questions
Are polynomials on the SAT?
Yes, as part of Advanced Math. Advanced Math makes up about 35% of SAT Math, roughly 13 to 15 questions.
What is the remainder theorem?
The remainder when p(x) is divided by (x − a) equals p(a).
How do I simplify rational expressions?
Factor the numerator and denominator completely, then cancel matching factors.