Expand products of polynomials, factor by common factor, grouping and special patterns, and simplify polynomial and rational expressions (Algebra).

What this topic is asking

The ACT tests expanding and factoring polynomial expressions, the two-way street between a product and its expanded form. Factoring also underpins solving quadratics and simplifying rational expressions, so the patterns here pay off across the test.

Expanding products

Distribution generalises FOIL to any product.

Factoring in order

Factoring is fastest when you check the patterns in a fixed sequence.

Skipping the GCF step is the most common reason an answer is "not fully factored".

The difference of squares

The pattern $a^{2} - b^{2} = (a - b)(a + b)$ is one of the most frequently tested on the ACT. It applies whenever you see a perfect square minus a perfect square: $x^{2} - 25$, $9x^{2} - 16$ (which is $(3x)^{2} - 4^{2} = (3x - 4)(3x + 4)$), or $x^{4} - 1$ (which factors further). A sum of squares like $x^{2} + 9$ does not factor over the real numbers, a distinction the ACT sometimes probes.

Factoring a general quadratic

For $ax^{2} + bx + c$, find two numbers that multiply to $a \cdot c$ and add to $b$, then split the middle term and factor by grouping. For $2x^{2} + 7x + 3$: $a \cdot c = 6$, and $6$ and $1$ multiply to 6 and add to 7, so $2x^{2} + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$. When $a = 1$, this reduces to the simple "multiply to $c$, add to $b$" method.

Simplifying rational expressions

A rational expression is a ratio of polynomials. Factor the numerator and denominator, then cancel common factors. For $\dfrac{x^{2} - 9}{x^{2} + 4x + 3}$, factor to $\dfrac{(x - 3)(x + 3)}{(x + 1)(x + 3)}$ and cancel $(x + 3)$ to get $\dfrac{x - 3}{x + 1}$ (for $x \neq -3$). You can only cancel factors, never individual terms across a sum.

Polynomial arithmetic and degree

Beyond factoring, the ACT may ask you to add, subtract or multiply polynomials and to read a polynomial's degree (its highest exponent). To add or subtract, combine like terms: $(3x^{2} + 2x - 1) + (x^{2} - 5x + 4) = 4x^{2} - 3x + 3$. To multiply a binomial by a trinomial, distribute every term of the first across the second and collect like terms. The degree of a product is the sum of the degrees of the factors, so a quadratic times a quadratic is degree 4. The leading coefficient and degree together control the end behaviour of the graph, a fact the Functions area builds on.

Why factoring is a high-yield skill

Factoring is the engine behind solving quadratics, simplifying fractions, and finding zeros of functions. Building the habit of "GCF first, then patterns, then quadratic" makes factoring fast and reliable, and expanding to check guarantees you have not introduced a sign error. Because these patterns recur throughout the Algebra and Functions areas, time spent here lifts your score broadly.

Try this

Q1. Expand $(x - 5)(x + 5)$. [1 point]

• Cue. Difference of squares: $x^{2} - 25$.

Q2. Factor completely $2x^{2} - 18$. [1 point]

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