Evaluate p(k) to find the remainder when dividing by (x−k), and check if (x−k) is a factor — no long division needed.
2x^3 - x + 5| k | p(k) | Is (x−k) a factor? |
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Remainder = p(k) when dividing by (x − k)
When a polynomial p(x) is divided by a linear factor (x − k), the remainder equals p(k) — simply evaluate the polynomial at x = k.
This replaces polynomial long division entirely when you only need the remainder. Plug k into every term, compute each, and sum them up.
Example: to find the remainder of p(x) = x³ − 4x² + x + 6 divided by (x − 3), compute p(3) = 27 − 36 + 3 + 6 = 0.
(x − k) is a factor of p(x) ⟺ p(k) = 0
The Factor Theorem is a direct consequence of the Remainder Theorem: (x − k) is a factor if and only if p(k) = 0.
Use it to quickly test rational roots — if p(k) = 0, then k is a zero of p(x) and (x − k) is a factor you can divide out (via synthetic or long division) to reduce the degree.
Combined with the Rational Root Theorem (which lists all possible rational k values), the Factor Theorem provides an efficient root-finding strategy for polynomials with integer coefficients.
One-on-one Algebra 2 tutoring makes the Remainder Theorem, Factor Theorem, and polynomial division click together — so you can tackle any root-finding problem with confidence.