Factor Special Products

Perfect squares and differences of squares. In MathBound, the free Algebra I adventure game, Butterfly Hinge asks these problems, and the only way past is to solve them.

Part of Polynomials and Factoring: chapter 8 of the standard course.

Play this battle free

10 questions, no sign-up, nothing saved. Climbs from the basics to harder questions.

What you will practice

Perfect Square Trinomials. Factor a perfect square trinomial with a positive middle term and leading coefficient of 1.
Differences of Squares. Factor a basic difference of two squares with a leading coefficient of 1.
Factor GCF First. Factor out an integer GCF before identifying and factoring a difference of squares.

Worked examples

Made by the game’s own question engine. Try each one, then open its steps.

Example 1 · Perfect Square Trinomials

What

goes

in

the

box?

x^{2} + 12 x + 36 = (x + 6) □

$(x + 5)$
$(x - 6)$
$(x + 6)$
$(x + 7)$

Show the worked steps

$36$ $is$ $6^{2}$ $and$ $the$ $middle$ $term$ $is$ $2 \cdot 1 \cdot 6 = 12 .$
$That$ $is$ $the$ $perfect$ $square$ $pattern:$ $(x + 6)^{2} .$
$Taking$ $out$ $(x + 6)$ $leaves$ $(x + 6) .$

Answer: $(x + 6)$

Example 2 · Differences of Squares

Solve

by

factoring:

25 x^{2} - 36 = 0

$x = \frac{- 6}{5}$ $or$ $x = \frac{6}{5}$
$x = \frac{- 36}{25}$ $or$ $x = \frac{36}{25}$
$x = \frac{6}{5}$
$x = \frac{- 6}{5}$

Show the worked steps

$Factor:$ $25 x^{2} - 36 = (5 x + 6) (5 x - 6) = 0.$
$Set$ $each$ $factor$ $to$ $zero:$ $x = \frac{- 6}{5}$ $or$ $x = \frac{6}{5} .$

Answer: $x = \frac{- 6}{5}$ $or$ $x = \frac{6}{5}$

Example 3 · Factor GCF First

Factor:

8 - 2 x^{2}

$2 (2 - x)^{2}$
$2 (x + 2) (x - 2)$
$(2 + x) (2 - x)$
$2 (2 + x) (2 - x)$

Show the worked steps

$Take$ $out$ $the$ $GCF$ $2$ $first.$
$4$ $is$ $2^{2}$ $and$ $x^{2}$ $is$ $(x)^{2} .$
$A$ $difference$ $of$ $squares$ $is$ $the$ $sum$ $times$ $the$ $difference:$ $(2 + x) (2 - x) .$

Answer: $2 (2 + x) (2 - x)$

Where it is in your textbook

Textbook	Lessons
Standard course	8.7
Big Ideas Math Algebra 1	7.7
Big Ideas Math Algebra 1 (2022)	7.7
CPM Core Connections Algebra	8.1.5, 8.2.5
CPS Skyline Algebra	7.3-7.5, 7.6-7.7
enVision Algebra 1	7-7
enVision Algebra 1 (2024)	7-7
Eureka Math Algebra I (EngageNY)	4.3-4.4, 4.11-4.13
Glencoe Algebra 1	8-8, 8-9
HMH Into Algebra 1	17.4
Holt McDougal Larson Algebra 1	8.7
Illustrative Mathematics Algebra 1	7.6-7.8, 7.11-7.14
Illustrative Mathematics Algebra 1 (v.360)	8.6-8.8, 8.11-8.14
Math Nation Algebra 1 (Florida B.E.S.T.)	7.7, 7.8, 8.7
Open Up NC Math 1 (MVP)	7.4
OpenStax Elementary Algebra 2e	7.4
Pearson Algebra 1 Common Core	8-7
Reveal Algebra 1	10-7
SpringBoard Algebra 1	26

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