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Using Structure to
Rewrite Expressions

Learn to see hidden patterns inside algebraic expressions: factor higher-degree polynomials, recognize difference of squares, and simplify complex expressions by treating chunks as single variables.

Algebra 2

What Does "Structure" Mean?

In algebra, structure refers to the way an expression is built from its parts: its terms, operations, and groupings. Recognizing structure means seeing that a complicated expression follows a familiar pattern you already know how to work with.

The Power of Substitution Thinking

The key technique is to let a chunk equal a new variable. If you see \(x^4 - y^4\), notice that \(x^4 = (x^2)^2\). Let \(u = x^2\) and the expression becomes \(u^2 - (y^2)^2\): a simple difference of squares.

\(x^4 - y^4\)
Let \(u = x^2\)
\(u^2 - (y^2)^2\)
Factor!
\(a^2 - b^2\)
 Rewrite the original so it matches a known pattern

Why Does This Matter?

Simplification
Complex expressions reduce to simpler factors, making equations easier to solve.
Solving Equations
Factored form reveals roots directly: set each factor equal to zero.
Future Math
Precalculus, calculus, and beyond all rely on recognizing algebraic structure.

The Core Patterns to Recognize

1
Difference of Squares: \(a^2 - b^2 = (a-b)(a+b)\)
2
Perfect Square Trinomial: \(a^2 \pm 2ab + b^2 = (a \pm b)^2\)
3
Sum/Difference of Cubes: \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
4
Grouping: When four terms share pairwise common factors.

Difference of Squares: Deep Dive

\(a^2\)
Side = \(a\)
 −
\(b^2\)
Side = \(b\)
 =
\((a-b)(a+b)\)
 Two linear factors

Difference vs. Sum of Squares

Difference of Squares Sum of Squares
Form \(a^2 - b^2\) \(a^2 + b^2\)
Factorable over \(\mathbb{R}\)? Yes No
Factored Form \((a-b)(a+b)\) Prime (irreducible)
Example \(x^2 - 9 = (x-3)(x+3)\) \(x^2 + 9\) cannot factor

How to Apply Structural Rewriting

1
Scan for perfect powers. Are any terms perfect squares, cubes, or higher? Rewrite them with exponents.
2
Identify a known pattern. Does the expression match \(a^2 - b^2\), \(a^2 \pm 2ab + b^2\), or \(a^3 \pm b^3\)?
3
Define your "chunks." Assign \(a = \ldots\) and \(b = \ldots\) explicitly. Write them down.
4
Apply the formula. Substitute your chunks into the factoring identity.
5
Check for further factoring. Can any resulting factor be factored again?

Example 1: \(x^4 - y^4\)

\(x^4 - y^4 = (x^2)^2 - (y^2)^2\)
Difference of squares: \(a = x^2,\ b = y^2\)
\((x^2 - y^2)(x^2 + y^2)\)
\((x - y)(x + y)(x^2 + y^2)\)

Example 2: \(9x^4 - 49\)

1
Rewrite as squares: \(9x^4 = (3x^2)^2\) and \(49 = 7^2\)
2
Identify pattern: \((3x^2)^2 - 7^2\) is a difference of squares with \(a = 3x^2\), \(b = 7\)
3
Factor: \((3x^2 - 7)(3x^2 + 7)\)
4
Check: Neither factor is a difference of squares, so we are done.

Example 3: \(x^6 - 64\)

This can be viewed two ways: as a difference of squares or a difference of cubes. Let's see both paths and which gives the complete factorization.

Path A: Squares First
\((x^3)^2 - 8^2\)
\(= (x^3 - 8)(x^3 + 8)\)
Each is a sum/diff of cubes!
\(= (x-2)(x^2+2x+4)\)
\(\cdot(x+2)(x^2-2x+4)\)
Path B: Cubes First
\((x^2)^3 - 4^3\)
\(= (x^2 - 4)(x^4 + 4x^2 + 16)\)
\(x^2 - 4 = (x-2)(x+2)\)
The term \(x^4+4x^2+16\) has a non-obvious factorization, but it leads to the same final answer.

Spotting Perfect Square Trinomials

A perfect square trinomial has the form \(a^2 + 2ab + b^2 = (a + b)^2\) or \(a^2 - 2ab + b^2 = (a - b)^2\). The middle term is always twice the product of the square roots of the first and last terms.

1
Check first and last terms: Are they perfect squares?
2
Check middle term: Does it equal \(2 \cdot \sqrt{\text{first}} \cdot \sqrt{\text{last}}\)?
3
Example: \(4x^2 + 12x + 9 = (2x)^2 + 2(2x)(3) + 3^2 = (2x + 3)^2\)

Factoring by Grouping

When an expression has four terms, try grouping them in pairs, factoring out a GCF from each pair, and then factoring out the common binomial.

\(x^3 + 3x^2 + 2x + 6\)
\(x^2(x + 3)\)
\(2(x + 3)\)
 +
\((x + 3)(x^2 + 2)\)

Sum and Difference of Cubes

Sum of Cubes
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Difference of Cubes
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Example: \(8x^3 - 27 = (2x)^3 - 3^3 = (2x - 3)(4x^2 + 6x + 9)\)

Common Mistakes

✕ Factoring a Sum of Squares
Writing \(x^2 + 9 = (x+3)(x-3)\) is wrong. \(a^2 + b^2\) does not factor over \(\mathbb{R}\).
✕ Forgetting to Factor Completely
Stopping at \((x^2 - 9)(x^2 + 9)\) and missing that \(x^2 - 9 = (x-3)(x+3)\).
✕ Sign Errors in Cubes
Mixing up the signs in \(a^3 - b^3\): the trinomial factor has \(+ab\), not \(-ab\).

Summary of Patterns

Pattern Form Factored Result
Diff. of Squares \(a^2 - b^2\) \((a-b)(a+b)\)
Perfect Sq. Trinomial \(a^2 \pm 2ab + b^2\) \((a \pm b)^2\)
Sum of Cubes \(a^3 + b^3\) \((a+b)(a^2 - ab + b^2)\)
Diff. of Cubes \(a^3 - b^3\) \((a-b)(a^2 + ab + b^2)\)
Grouping 4 terms with pairs Factor each pair, then common binomial

Question 1 of 5

Which of the following is the complete factorization of \(x^4 - 16\)?

Question 2 of 5

Factor completely: \(25x^2 - 40x + 16\)

Question 3 of 5

Which expression is equivalent to \(27a^3 + 8b^3\)?

Question 4 of 5

Factor by grouping: \(x^3 - 5x^2 + 4x - 20\)

Question 5 of 5

What structural pattern does the expression \(16x^4 - 81y^4\) follow at the first factoring step?