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SAT Math: The Remainder Theorem in Depth
22+ practice questions in Praczo
What you need to know
The concept, explained
- 1
The Remainder Theorem states that if you divide a polynomial p(x) by a linear factor (x - c), the remainder is precisely p(c).
- 2
If the remainder is 0, it means (x - c) is a PERFECT FACTOR of p(x), and c is a root/zero of the equation.
- 3
If the SAT says "p(x) is divisible by (x+3)", that means the remainder is 0, so p(-3) = 0.
- 4
Conversely, if told p(4) = 7, it means dividing p(x) by (x - 4) leaves a remainder of 7.
Common mistakes
- ✗ Plugging in the wrong sign (e.g., plugging in 2 when dividing by x + 2, instead of -2).
- ✗ Attempting full polynomial long division incorrectly when simply plugging the number conceptually solves it in ten seconds.
Try a sample question
SAT-style practice
If the polynomial P(x) = x³ - kx² + x + 6 is divisible by (x - 2), what is the value of k?
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