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SAT Math: Polynomial Remainder Theorem
22+ practice questions in Praczo
What you need to know
The concept, explained
- 1
Remainder Theorem: when polynomial p(x) is divided by (x − a), the remainder equals p(a).
- 2
Factor Theorem (corollary): (x − a) is a factor of p(x) if and only if p(a) = 0.
- 3
To check if (x − 2) is a factor, evaluate p(2). If p(2) = 0, it is a factor.
- 4
The SAT uses this to ask for the value of a constant in a polynomial given that a certain binomial is a factor.
- 5
You do not need to perform long division — just substitute the zero of the divisor.
Common mistakes
- ✗ Substituting the wrong value: for (x − 3), substitute x = 3, not x = −3.
- ✗ Confusing remainder with quotient. The remainder theorem gives the remainder, not the full result of the division.
Try a sample question
SAT-style practice
If p(x) = x³ − 4x² + kx − 6 and (x − 3) is a factor of p(x), what is the value of k?
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