Equivalent Expressions

Equivalent expressions questions ask you to rewrite algebraic expressions in different but mathematically identical forms. This is one of the most frequently tested topics in the Advanced Math domain on the Digital SAT, typically appearing in 3–5 questions per test. The core skill is recognizing structure: can you factor a polynomial, simplify a fraction containing variables, or convert between exponent and radical notation? You won't always need to solve for a variable. Often the question simply asks, "Which expression is equivalent to...?" and your job is to manipulate the original into one of the answer choices.

Factoring Out a Common Factor

The simplest form of factoring is pulling out a greatest common factor (GCF) shared by every term. Before attempting any other technique, always scan for a GCF first.

Example: Which expression is equivalent to $6x^3 + 15x^2 - 9x$?

Every term contains a factor of $3x$:

$6x^3 + 15x^2 - 9x = 3x(2x^2 + 5x - 3)$

That's it. On the SAT, one of the answer choices will match $3x(2x^2 + 5x - 3)$. If the choices show further factoring, you'd continue (see the trinomial section below), but always start here.

Common mistake: Forgetting to factor the GCF from every term. If you pull $3x$ out of the first two terms but miss the last one, your factored form won't multiply back to the original. Always verify by distributing your answer back out.

Factoring Trinomials and Binomials

Simple trinomials (leading coefficient of 1)

For polynomials of the form $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. Those numbers go directly into two binomials.

Example: Factor $x^2 - 7x + 12$

You need two numbers that multiply to $12$ and add to $-7$. Those are $-3$ and $-4$.

$x^2 - 7x + 12 = (x - 3)(x - 4)$

Trinomials with a leading coefficient other than 1

When the expression looks like $ax^2 + bx + c$ with $a \neq 1$, use the AC method:

1. Multiply $a \times c$
2. Find two numbers that multiply to $ac$ and add to $b$
3. Split the middle term and factor by grouping

Example: Factor $2x^2 + 7x + 3$

$a \times c = 2 \times 3 = 6$

Two numbers that multiply to $6$ and add to $7$: that's $1$ and $6$.

$2x^2 + 1x + 6x + 3$

Group: $(2x^2 + x) + (6x + 3) = x(2x + 1) + 3(2x + 1)$

Factor out the common binomial: $(2x + 1)(x + 3)$

Difference of squares

Any expression of the form $a^2 - b^2$ factors into $(a + b)(a - b)$. Both terms must be perfect squares separated by subtraction.

Example: Factor $9x^2 - 49$

Recognize $(3x)^2 - (7)^2$, so:

$9x^2 - 49 = (3x + 7)(3x - 7)$

Watch for disguised versions. The expression $x^4 - 16$ is $(x^2)^2 - 4^2$, which factors to $(x^2 + 4)(x^2 - 4)$. Then $x^2 - 4$ is itself a difference of squares: $(x + 2)(x - 2)$. The SAT loves these layered problems.

Important: A sum of squares like $x^2 + 25$ does not factor over the real numbers. If you see it in an answer choice as $(x + 5)(x + 5)$, that's wrong because $(x + 5)^2 = x^2 + 10x + 25$, not $x^2 + 25$.

Adding, Subtracting, and Multiplying Polynomials

These questions test whether you can fluently expand and combine polynomials. The SAT expects speed and accuracy here.

Multiplying polynomials

Distribute every term in the first factor to every term in the second, then combine like terms.

Example: Expand $(3x + 2)(x^2 - 4x + 5)$

$3x \cdot x^2 = 3x^3$ $3x \cdot (-4x) = -12x^2$ $3x \cdot 5 = 15x$ $2 \cdot x^2 = 2x^2$ $2 \cdot (-4x) = -8x$ $2 \cdot 5 = 10$

Combine like terms:

$3x^3 + (-12x^2 + 2x^2) + (15x - 8x) + 10 = 3x^3 - 10x^2 + 7x + 10$

Adding and subtracting polynomials

Combine like terms carefully. The most common trap is sign errors when subtracting, because you must distribute the negative to every term in the second polynomial.

Example: Simplify $(4x^2 + 3x - 1) - (2x^2 - 5x + 7)$

Distribute the negative sign:

$4x^2 + 3x - 1 - 2x^2 + 5x - 7$

Combine: $2x^2 + 8x - 8$

Forgetting to flip the sign on $-5x$ to $+5x$ or on $+7$ to $-7$ is exactly the kind of error that produces a wrong answer choice on the SAT. Those trap answers are there on purpose.

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. To simplify, factor both top and bottom, then cancel common factors.

Example: Simplify $\frac{x^2 - 9}{x^2 + 5x + 6}$

Factor the numerator (difference of squares): $x^2 - 9 = (x + 3)(x - 3)$

Factor the denominator (trinomial): $x^2 + 5x + 6 = (x + 3)(x + 2)$

$\frac{(x + 3)(x - 3)}{(x + 3)(x + 2)} = \frac{x - 3}{x + 2}$

This is valid for $x \neq -3$ (since that would make the original denominator zero). The SAT sometimes includes the restriction in the question stem: "for $x \neq -3$, which expression is equivalent to..."

Harder example: Simplify $\frac{2x^2 - 8}{x^2 - 4x + 4}$

Numerator: $2x^2 - 8 = 2(x^2 - 4) = 2(x + 2)(x - 2)$

Denominator: $x^2 - 4x + 4 = (x - 2)^2$

$\frac{2(x + 2)(x - 2)}{(x - 2)^2} = \frac{2(x + 2)}{x - 2}$

Notice how the GCF step in the numerator was essential. Without pulling out the 2 first, you might not see the difference of squares hiding inside.

Rational Exponents and Radical Form

The SAT tests whether you can convert between exponent notation and radical form using this rule:

$x^{\frac{m}{n}} = \sqrt[n]{x^m}$

The denominator of the fractional exponent becomes the index of the radical. The numerator stays as the power.

Example: Rewrite $x^{\frac{3}{4}}$ in radical form.

$x^{\frac{3}{4}} = \sqrt[4]{x^3}$

Example going the other direction: Which expression is equivalent to $\sqrt[3]{x^5}$?

$\sqrt[3]{x^5} = x^{\frac{5}{3}}$

The SAT might also combine this with simplification:

Example: Simplify $\frac{x^{\frac{1}{2}} \cdot x^{\frac{2}{3}}}{x^{\frac{1}{6}}}$

Add exponents in the numerator: $x^{\frac{1}{2} + \frac{2}{3}} = x^{\frac{3}{6} + \frac{4}{6}} = x^{\frac{7}{6}}$

Subtract the denominator's exponent: $x^{\frac{7}{6} - \frac{1}{6}} = x^{\frac{6}{6}} = x^1 = x$

These problems reward comfort with fraction arithmetic as much as they test exponent rules.

What to Watch For on Test Day

• Factor the GCF first, always. Many problems become dramatically simpler once you pull out a common factor. It also prevents you from missing a factoring that the answer choices expect.
• Check your signs when subtracting polynomials. The SAT deliberately places answer choices that match common sign errors. Distribute the negative to every term before combining.
• Don't cancel terms in rational expressions; cancel factors. You cannot simplify $\frac{x^2 + 3}{x}$ by "canceling" the $x$. You can only cancel factors that multiply across the entire numerator and denominator. Factor first, then cancel.
• Recognize structure before computing. If a question shows $4x^2 - 25$, you should immediately see a difference of squares rather than trying to use the AC method. Pattern recognition saves significant time.
• Convert fractional exponents to radicals (or vice versa) based on the answer choices. If the choices are in radical form, convert your work to match. The denominator of the exponent is the root index; the numerator is the power.