Remainder and Factor Theorems

Algebra 2 · Axiom Academy

LESSON Remainder and Factor Theorems Dividing P(x) by (x-a) leaves remainder P(a) — and (x-a) is a factor exactly when P(a)=0 . Dividing P(x) by (x-a) works exactly like dividing whole numbers: you get a quotient q(x) and a remainder . Because the divisor (x-a) has degree 1 , that remainder can only be a constant r — it has no room left for an x . So every such division can be written as one clean identity. dividend = divisor quotient + remainder Watch synthetic division run on , where a = 2 : bring down the leading coefficient, multiply by 2 , add down each column. The bottom row hands you the quotient and the remainder. The proof falls straight out of the division identity. Watch the animation set x=a and collapse the quotient term to nothing. Substitute x=a : the (a-a) factor is zero …so the remainder r is the value P(a) For P(x) = x^2 + 3x + 5 at a = 2 , evaluate directly: That is the very same 15 we got as the remainder from the division — no long division required. This is the Remainder Theorem's corollary. The remainder is P(a) , so a remainder of 0 means (x-a) divides P(x) evenly — a factor. And the reverse holds too: if (x-a) is a factor, the division leaves nothing, so P(a)=0 . The remainder is nonzero, the graph misses the axis at x=a , and (x-a) is not a factor. The remainder vanishes, the graph crosses the axis at x=a , and (x-a) is a factor.

This is the written version of the interactive lesson above. See the full Algebra 2 course.