SAT — mathMedium5 min read

Factoring Special Forms

Master factoring difference of squares and perfect square trinomials for the Digital SAT. Recognise patterns for fast, accurate factoring.

Tutor AI Team2026-02-25T01:20:08.788Z

Certain polynomial expressions have special structures that allow them to be factored instantly — without trial and error. The Digital SAT rewards students who can recognise these patterns quickly. The two key patterns are the difference of two squares and perfect square trinomials.

Core Concepts

Difference of Two Squares (DOTS)

$a^2 - b^2 = (a + b)(a - b)$

This pattern applies when:

There are exactly two terms
They are separated by a minus sign
Both terms are perfect squares

Examples: $x^2 - 49 = (x + 7)(x - 7)$ $4x^2 - 25 = (2x + 5)(2x - 5)$ $9x^2 - 16y^2 = (3x + 4y)(3x - 4y)$

Important: $a^2 + b^2$ (sum of squares) does NOT factor over the real numbers.

Perfect Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$

Recognise this pattern by checking:

First term is a perfect square
Last term is a perfect square
Middle term = $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$

Examples: $x^2 + 10x + 25 = (x + 5)^2$ $x^2 - 14x + 49 = (x - 7)^2$ $4x^2 + 12x + 9 = (2x + 3)^2$

Check: $2 \times 2x \times 3 = 12x$ ✓

Factoring in Multiple Steps

Always look for a common factor first, then check for special forms.

$3x^2 - 48 = 3(x^2 - 16) = 3(x + 4)(x - 4)$

$2x^2 + 12x + 18 = 2(x^2 + 6x + 9) = 2(x + 3)^2$

Sum and Difference of Cubes (Less Common on SAT)

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

These appear rarely on the SAT but are good to know.

Strategy Tips

Tip 1: Scan for Two Terms with a Minus

If you see $\text{something}^2 - \text{something}^2$ , it's DOTS. Factor immediately.

Tip 2: Check the Middle Term for Perfect Squares

For trinomials, check if the middle term equals $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$ .

Tip 3: Factor Out the GCF First

Before looking for special forms, always check for a greatest common factor.

Tip 4: Recognise Disguised Patterns

$(x+1)^2 - 9 = [(x+1) + 3][(x+1) - 3] = (x+4)(x-2)$

The " $a$ " in DOTS can be an expression, not just a single term.

Tip 5: Verify by Expanding

Quickly expand your factored form to check it matches the original.

Worked Example: Example 1

Problem

Factor $x^2 - 64$ .

$= (x + 8)(x - 8)$

Solution

Worked Example: Example 2

Problem

Factor $25x^2 - 1$ .

$= (5x + 1)(5x - 1)$

Solution

Worked Example: Example 3

Problem

Factor $x^2 + 16x + 64$ .

Check: $\sqrt{x^2} = x$ , $\sqrt{64} = 8$ , $2(x)(8) = 16x$ ✓

$= (x + 8)^2$

Solution

Worked Example: SAT-Style

Problem

Which of the following is equivalent to $(x + 3)^2 - 16$ ?

This is DOTS with $a = (x+3)$ and $b = 4$ :

$= [(x+3) + 4][(x+3) - 4] = (x+7)(x-1)$

Solution

Worked Example: Example 5

Problem

Factor completely: $5x^3 - 45x$ .

$= 5x(x^2 - 9) = 5x(x+3)(x-3)$

Solution

Practice Problems

Problem 1

Factor $x^2 - 100$ .

Problem 2

Factor $9x^2 - 4$ .

Problem 3

Factor $x^2 - 6x + 9$ .

Problem 4

Factor completely $2x^2 - 50$ .

Problem 5

Which expression is equivalent to $(2x - 1)^2 - 25$ ?

Problem 6

Is $x^2 + 9$ factorable over the real numbers?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Trying to factor $a^2 + b^2$ . The sum of squares does NOT factor.
Missing the GCF. Always extract common factors first.
Incorrect perfect square check. Verify that the middle term is exactly $2ab$ .
Not recognising disguised DOTS. $(x+y)^2 - z^2$ is still a difference of squares.
Stopping too early. After factoring out GCF, check if the remaining expression factors further.

Frequently Asked Questions

How do I quickly check if a number is a perfect square?

Know your squares: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$ . For variables: $x^2, x^4, x^6$ are perfect squares.

Does the SAT test sum of cubes?

Rarely, but it's possible. Focus on DOTS and perfect square trinomials first.

Can DOTS be applied to higher powers?

Yes! $x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)$ .

What if the DOTS has coefficients?

Same pattern: $16x^2 - 49 = (4x)^2 - 7^2 = (4x + 7)(4x - 7)$ .

How is this tested on the SAT?

"Which is equivalent to..." or "Factor completely." Also appears in solving equations.

Key Takeaways

✓
DOTS: $a^2 - b^2 = (a + b)(a - b)$ . Look for two squares separated by minus.
✓
Perfect square trinomial: $a^2 ± 2ab + b^2 = (a ± b)^2$ . Check the middle term.
✓
$a^2 + b^2$ does NOT factor over the reals.
✓
GCF first — always check for a common factor before applying special forms.
✓
" $a$ " can be an expression — look for disguised patterns.
✓
Verify by expanding your factored form.

Tags:

#sat#math#advanced-math#factoring#special-forms

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Core Concepts

Difference of Two Squares (DOTS)

Perfect Square Trinomials

Factoring in Multiple Steps

Sum and Difference of Cubes (Less Common on SAT)

Strategy Tips

Tip 1: Scan for Two Terms with a Minus

Tip 2: Check the Middle Term for Perfect Squares

Tip 3: Factor Out the GCF First

Tip 4: Recognise Disguised Patterns

Tip 5: Verify by Expanding

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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