Factor polynomials using common factors and standard patterns, and identify the zeros of a polynomial from its factored form (TN A1.A.SSE.A.2, A1.A.APR.A.3).

What this topic is asking

This page combines two standards: A1.A.SSE.A.2 (rewrite an expression by its structure) and A1.A.APR.A.3 (identify the zeros of a polynomial from a suitable factorization and use them to sketch the graph). Factoring is the structural rewrite; the zeros are what the factored form reveals. On TNReady, factoring is tested both as a stand-alone skill and as the route to a function's $x$-intercepts.

A factoring routine

Use the same sequence every time.

1. GCF. Factor out the greatest common factor of all terms first.
2. Count terms. Two terms means try difference of squares; three terms means factor a trinomial.
3. Trinomial with leading coefficient 1. Find two numbers that multiply to the constant and add to the linear coefficient.
4. Trinomial with leading coefficient $\ne 1$. Factor by grouping (the "$ac$ method").
5. Check. Multiply the factors back; the product should be the original.

Factoring a simple trinomial

For $x^2 + bx + c$, you need a number pair with product $c$ and sum $b$. The signs of $b$ and $c$ narrow the search: if $c > 0$ the numbers share $b$'s sign; if $c < 0$ they have opposite signs.

From factors to zeros

Standard A1.A.APR.A.3 is about reading the zeros off the factored form. If $f(x) = (x + 3)(x - 4)$, then $f(x) = 0$ exactly when one factor is zero, by the zero-product property. So $x = -3$ or $x = 4$, and these are the $x$-intercepts of the parabola. Plotting the zeros, together with the vertex halfway between them, sketches the graph, which is how the standard links factoring to graphing.

How TNReady examines this topic

• Equation response. Type the fully factored form, scored by exact match.
• Multiple choice. Choose the factorization, or the zeros of a graphed function, with sign-flip distractors.
• Graphing. Plot the $x$-intercepts of a function from its factors.

A clarifying idea is that the zeros are the opposite sign of the constants inside the factors: $(x + 3)$ gives a zero at $x = -3$, not $x = +3$. Reversing this sign is the single most common error, so set each factor equal to zero rather than reading the sign directly.

Choosing the right pattern quickly

Speed on this category comes from recognizing the form before you start. A glance at the term count and signs usually decides the method. Two terms that are both perfect squares with a minus between them is a difference of squares, so reach straight for $(a - b)(a + b)$. A three-term expression whose leading coefficient is $1$ is a simple trinomial, so search for the product-and-sum pair. A three-term expression with a leading coefficient larger than $1$ calls for the $ac$ method and grouping. And any expression at all gets a GCF check first. Training this quick triage means you spend your time executing a method rather than guessing which one applies, which is exactly what the timed Subpart 1 rewards.

Why factor completely, and what does not factor

"Completely" means no factor can be broken down further. After pulling a GCF, always re-check the remaining factor: $2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)$, where stopping at $2(x^2 - 4)$ leaves a difference of squares unfactored. Not every quadratic factors over the integers, though. If no integer pair multiplies to $ac$ and adds to $b$, the trinomial is prime over the integers, and you would solve it with the quadratic formula instead. Recognizing a prime trinomial quickly, rather than hunting forever for factors that do not exist, saves time on the test.

Try this

Q1. Factor $x^2 - 9x + 20$. [1 point]

• Cue. $-4$ and $-5$ multiply to $20$ and add to $-9$: $(x - 4)(x - 5)$.

Q2. Factor $3x^2 - 12$ completely, then state its zeros. [2 points]

• Cue. $3(x^2 - 4) = 3(x - 2)(x + 2)$; zeros at $x = 2$ and $x = -2$.