Factor Theorem

SAT Glossary

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.

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is just $p(3)=0$.","section":"Mathematics","source":"KomFi SAT vF - Chapter 35: Equivalent Expressions and Polynomial Algebra","necessity":4,"learnMore":"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.","aliases":[],"topic":"Math","priority":"specialized","learningStage":"intermediate","prerequisiteIds":["Composition of functions","Pythagorean identity"],"questionCount":2,"id":"Factor Theorem","courseId":"sat","publicSeo":true,"seoSlug":"factor-theorem-cplu66"}

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Factor Theorem

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