Factoring Common Factors and Special Products

5.2: Factoring Common Factors and Special Products. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 2 of 63 parts Practice Flashcards

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Common-factor extraction

Objective: Factor out the greatest common numerical and variable factor.

Prerequisite skills: polynomial multiplication and integer factors. The greatest common factor is the largest product that divides every term. Divide each term by it to determine the remaining factor, then distribute to check. Factoring changes representation, not value. Common error: leaving one term undivided after extracting a factor. Restrictions and interpretation: Factorization must reproduce every original term; the greatest common factor includes shared variable powers.

Formulas

$ab + ac = a(b + c)$

Worked examples

Try itWhat is the greatest positive integer common factor of 138x² and 207x?Show answer

The coefficients are 6(23) and 9(23). Since gcd(6,9)=3, their greatest common numerical factor is 3(23) = 69. The full monomial GCF also contains x. Therefore the requested answer is 69.

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Common-factor extraction

Objective: Factor out the greatest common numerical and variable factor.

Formulas

$ab + ac = a(b + c)$

Worked examples

Try itWhat is the greatest positive integer common factor of 138x² and 207x?Show answer

The coefficients are 6(23) and 9(23). Since gcd(6,9)=3, their greatest common numerical factor is 3(23) = 69. The full monomial GCF also contains x. Therefore the requested answer is 69.

Difference of squares

Objective: Recognize and factor a difference of two squares.

A² − B² factors as (A − B)(A + B). The cross terms cancel on multiplication. A sum of squares does not factor this way over the real numbers. Common error: using two identical minus factors for a difference of squares. Restrictions and interpretation: Factorization must reproduce every original term; the greatest common factor includes shared variable powers.

Formulas

$A² − B² = (A − B)(A + B)$

Worked examples

Try itFor x² − 625 = (x − k)(x + k) with k positive, what is k?Show answer

Recognize 625 = 25². Therefore x² − 25² = (x − 25)(x + 25), so positive k = 25. Therefore the requested answer is 25.

Perfect-square structure

Objective: Recover a missing coefficient in a perfect-square trinomial.

A perfect-square trinomial has a first square, a last square, and a middle term twice the product of their square roots. Check the middle sign to determine whether the binomial uses + or −. Common error: omitting the cross term when squaring a sum. Restrictions and interpretation: Factorization must reproduce every original term; the greatest common factor includes shared variable powers. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$(ax + b)² = a²x² + 2abx + b²$

Worked examples

Try itIf 9x² + kx + 625 = (3x + 25)² for every x, what is k?Show answer

Expand (3x + 25)²: 9x² + 2(3x)(25) + 625. Thus the middle coefficient k is 6(25) = 150. Therefore the requested answer is 150.

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