Use the structure of an expression to identify ways to rewrite it, and write an equivalent factored form of a quadratic to reveal zeros (NC.M1.A-SSE.2, A-SSE.3).

What this topic is asking

Two standards work together. NC.M1.A-SSE.2 asks you to use the structure of an expression to see how it can be rewritten: spotting a common factor, a difference of squares, or a perfect-square pattern. NC.M1.A-SSE.3 narrows this to quadratics: write an equivalent factored form of $ax^2 + bx + c$ to reveal the zeros of the related equation or function. The theme is that equivalent forms are not equally useful, and the right rewrite exposes information you want.

The structures you must recognize

A-SSE.2 is a pattern-recognition standard. These three patterns cover almost every NC Math 1 item.

Recognizing the pattern is faster than blind trial. If you see $x^2 - 16$, the structure (square minus square) tells you to write $(x - 4)(x + 4)$ at once.

Using structure on a difference of squares

The cancelling middle terms are the signature of a difference of squares: when you multiply $(a - b)(a + b)$, the $+ab$ and $-ab$ always cancel.

Factored form reveals the zeros

A-SSE.3 is about choosing the form that shows the solutions. A quadratic in factored form displays its zeros directly.

If $y = (x - r)(x - s)$, then $y = 0$ exactly when one factor is zero, so $x = r$ or $x = s$. The zeros are also the $x$-intercepts of the parabola. This is why factoring is the first tool for solving quadratic equations and for graphing.

For example, $y = x^2 - x - 6 = (x - 3)(x + 2)$ has zeros $x = 3$ and $x = -2$, so the parabola crosses the $x$-axis at $(3, 0)$ and $(-2, 0)$.

When to factor out a GCF first

Always check for a common factor before applying another pattern, because pulling out the GCF can expose a simpler structure.

For $2x^2 - 8$, factor the GCF first: $2(x^2 - 4)$. Now $x^2 - 4$ is a difference of squares, so the full factorization is $2(x - 2)(x + 2)$. Skipping the GCF step would leave the difference of squares hidden.

How the NC Math 1 EOC examines this topic

• Multiple choice. Choose the equivalent factored form, or identify which factoring pattern applies.
• Gridded response. State a zero of a function after factoring, or enter a missing factor.
• Technology-enhanced. Match each expression to its factored form, or select all expressions equivalent to a given one.

A useful exam habit is to expand to check. Because the test scores factored answers by exact match in many formats, multiplying your factors back out catches sign errors before they cost a point.

Why equivalent forms carry different information

The same number can be written as $12$, $3 \times 4$, or $2^2 \times 3$, and which you choose depends on what you want to see. Expressions are the same. The expanded form $x^2 + 2x - 15$ makes the leading coefficient and constant obvious; the factored form $(x + 5)(x - 3)$ makes the zeros obvious; later courses add vertex form to make the turning point obvious. A-SSE asks you to move fluently between forms and to pick the one that answers the question, which is a habit that pays off through every Functions item on the test.

Try this

Q1. Factor $x^2 - 81$. [1 point]

• Cue. Difference of squares: $(x - 9)(x + 9)$.

Q2. Write $5x^2 + 15x$ in factored form and state the zeros of $y = 5x^2 + 15x$. [2 points]

• Cue. GCF: $5x(x + 3)$; zeros $x = 0$ and $x = -3$.