Factor, if possible, trinomials with real factors in the form $ax^2 + bx + c$, including perfect-square trinomials, and decide if a binomial is a difference of two squares and rewrite it (TEKS A.10E, A.10F).

What this topic is asking

Factoring is the bridge from polynomial arithmetic to solving quadratics, and TEKS A.10E and A.10F sit in the Number and Algebraic Methods category. STAAR expects you to factor trinomials of the form $ax^2 + bx + c$ with real factors (including perfect squares) and to recognize and rewrite a difference of two squares. The same skill reappears in the Quadratic Functions and Equations category when you solve $ax^2 + bx + c = 0$ by factoring, so this is high-value work.

Step one is always the GCF

Before any pattern, pull out the greatest common factor of all terms. It is the largest number and variable power dividing every term.

Skipping the GCF is the top reason a "complete factorization" answer is wrong: $4x^2 - 36$ must become $4(x^2 - 9)$ before the difference of squares is applied. On a multiple-choice item, a partially factored choice like $4(x^2 - 9)$ is a deliberate distractor.

Difference of two squares

A binomial that is one square minus another square factors by the reference-sheet identity:

A sum of squares such as $x^2 + 9$ does not factor over the real numbers, so do not force it. Recognize the squares: $49$, $36x^2$, $\frac{1}{4}$, and so on.

Perfect-square trinomials

When a trinomial is the square of a binomial, the reference sheet gives the pattern:

The test is whether the first and last terms are perfect squares and the middle term is twice their roots. For $x^2 + 10x + 25$: $\sqrt{x^2} = x$, $\sqrt{25} = 5$, and $2 \cdot x \cdot 5 = 10x$ matches, so it is $(x + 5)^2$.

Trinomials with a leading coefficient

For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$. For $ax^2 + bx + c$ with $a \neq 1$, use the AC method.

How STAAR examines factoring

• Multiple choice and multiselect. Pick the complete factorization, or select all expressions equivalent to a given polynomial. The GCF and the "stop too early" distractors are standard.
• Drag and drop. Drag the correct constants into binomial templates such as $(x + \square)(x + \square)$, which tests the product-and-sum reasoning directly.
• Inside quadratic items. Factoring is the fastest route to the zeros of a parabola and the solutions of $ax^2 + bx + c = 0$, so it threads through Reporting Category 4.

A clarifying check is that factoring undoes multiplying: expanding your factors must rebuild the original polynomial. If it does not, a sign or a number is off.

Reading the signs of a trinomial

The signs in $x^2 + bx + c$ tell you the signs of the factors before you search. If $c$ is positive, the two numbers share a sign (both positive when $b$ is positive, both negative when $b$ is negative), as in $x^2 - 7x + 12 = (x - 3)(x - 4)$. If $c$ is negative, the two numbers have opposite signs, and the larger absolute value carries the sign of $b$, as in $x^2 - x - 12 = (x - 4)(x + 3)$. Using the sign rule first narrows the candidate pairs immediately and prevents the common error of producing factors whose product has the wrong constant sign. Combined with the GCF-first habit and the two reference-sheet identities, this gives a complete, ordered procedure that handles every factorable polynomial the test puts in front of you.

Try this

Q1. Factor completely: $3x^2 - 27$. [1 point]

• Cue. GCF first: $3(x^2 - 9) = 3(x - 3)(x + 3)$.

Q2. Factor: $x^2 - 12x + 36$. [1 point]

• Cue. Perfect square: $\sqrt{36} = 6$ and $2 \cdot 6 = 12$, so $(x - 6)^2$.