Factor Theorem
SAT glossary · Math
Definition
The special case of the Remainder Theorem: (x-a) is a factor of p(x) if and only if p(a)=0. Run backwards, it answers the common parameter variant — 'for what k is x-3 a factor of x^3-4x^2+kx+5' is just p(3)=0.
Factor Theorem in plain English
The Factor Theorem is a shortcut version of a bigger idea (the Remainder Theorem): it says (x-a) is a factor of a polynomial p(x) exactly when plugging a into the polynomial gives you zero, i.e. p(a)=0. You can also run this idea backwards to solve for an unknown constant: if a question says 'x-3 is a factor of x^3-4x^2+kx+5, find k,' all you have to do is plug in x=3, set the whole expression equal to 0, and solve for k — you never need to do polynomial long division at all.
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