Lesson · Factoring and invalid operations

Factoring and invalid operations

Publisher: Anannt EducationReviewer: Precalculus Academic Lead (designated)Last review: 10 September 2026~28 min

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Retrieve

Warm up the algebra this lesson needs: expanding a difference of squares and extracting a common factor.

exactsymbolicfoundationalFamily fam-expand-binomialNo calculator

Expand

(x-4)(x+4)

Answer

Exact form required on this item.

exactsymbolicfoundationalFamily fam-gcfNo calculator

Factor out the greatest common factor:

6x^2-9x

Answer

Exact form required on this item.

Editorial blocks

Why this works

Factoring rewrites a polynomial as a product. Products are zero precisely when a factor is zero, and common factors of a fraction can be cancelled because they contribute a factor of

1

wherever they are nonzero.

Common mistake

Cancelling a letter that appears as a *term* (for example, turning

\dfrac{x+2}{x}

into

2

). A letter must be a factor of the whole numerator before it cancels.

Check your result

Expand your factors. Substitute a number that is not a root. If you cancelled, substitute a nearby value and confirm the original and simplified expressions agree there.

Explain your thinking

State whether a common factor was cancelled, and name the input that remains excluded.

Exam application

AP Precalculus Unit 1 expects reliable algebra before rational-function analysis. Invalid cancellation is a frequent source of lost domain restrictions.

Compact summary

Factor by reversing expansion. Cancel only common factors of an entire numerator and denominator. After cancelling a factor that was zero at an input, keep that input excluded from the domain.

Calculator guidance: No calculator. Factoring and cancellation are algebraic statements; a numerical check can confirm a candidate factorisation but cannot justify cancelling a term that is not a factor.

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After this lesson

Stay honest with the independent check if you can. When you are done — stuck or not — tell us where you are. Parent WhatsApp is required on the next page. Under 13: a parent should finish that form.

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