Equivalent Expressions: Free SAT Math Lesson
"Equivalent expressions" is one of the four officially named question types in College Board's Advanced Math domain, and it covers a skill you'll use inside nearly every other Advanced Math problem: rewriting an algebraic expression into a different, equally valid form. This free, complete lesson covers expanding and multiplying polynomials, factoring using the greatest common factor, difference of squares, and trinomial methods, simplifying rational expressions, and recognizing which expressions are algebraically equivalent, the SAT's signature "which of the following is equivalent to..." question format. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. Expanding and Multiplying Polynomials
To multiply two binomials, distribute each term of the first factor across every term of the second, often remembered as FOIL: First, Outer, Inner, Last. Two patterns are worth memorizing since they appear constantly: (a + b)² = a² + 2ab + b², and (a − b)(a + b) = a² − b², the difference of squares.
Worked Example: Expand (2x + 3)(x − 5).
2x(x) + 2x(−5) + 3(x) + 3(−5)
2x² − 10x + 3x − 15
2x² − 7x − 15
1. Expand: (x + 4)(x − 6)
Explanation: x² − 6x + 4x − 24 = x² − 2x − 24.
2. Expand: (3x − 2)(2x + 5)
Explanation: 6x² + 15x − 4x − 10 = 6x² + 11x − 10.
3. Expand: (x + 7)²
Explanation: Using (a + b)² = a² + 2ab + b² with a = x and b = 7: x² + 2(x)(7) + 49 = x² + 14x + 49.
4. Expand: (2x − 3)²
Explanation: Using (a − b)² = a² − 2ab + b² with a = 2x and b = 3: 4x² − 2(2x)(3) + 9 = 4x² − 12x + 9.
5. Expand: (x − 5)(x + 5)
Explanation: This matches the difference of squares pattern (a − b)(a + b) = a² − b²: x² − 25.
2. Factoring Expressions
Factoring reverses expanding. Three patterns to check, roughly in order: first pull out a greatest common factor (GCF) if one exists, then check whether the expression matches a difference of squares, a2 − b2 = (a − b)(a + b), or a perfect square trinomial, a² ± 2ab + b² = (a ± b)², and finally try factoring a general trinomial by finding two numbers with the right product and sum.
Worked Example: Factor 3x² + 15x completely.
Both terms share a factor of 3x.
3x² + 15x = 3x(x + 5)
1. Factor completely: 6x² − 9x
Explanation: The greatest common factor of 6x² and 9x is 3x: 6x² − 9x = 3x(2x − 3).
2. Factor: x² − 49
Explanation: This is a difference of squares: x² − 49 = x² − 7² = (x − 7)(x + 7).
3. Factor: x² + 10x + 25
Explanation: This matches a² + 2ab + b² with a = x and b = 5, since 2(x)(5) = 10x, so it factors as (x + 5)².
4. Factor: x² − 3x − 18
Explanation: Find two numbers that multiply to −18 and add to −3: −6 and 3. This gives (x − 6)(x + 3).
5. Factor: 4x² − 25
Explanation: This is a difference of squares: 4x² − 25 = (2x)² − 5² = (2x − 5)(2x + 5).
3. Simplifying Rational Expressions
A rational expression is a fraction with a polynomial in the numerator, denominator, or both. To simplify one, factor the numerator and denominator completely, then cancel any factors they share. A rational expression is undefined for any x-value that makes its denominator zero.
Worked Example: Simplify x² − 9x + 3.
(x − 3)(x + 3)x + 3 = x − 3, where x ≠ −3
1. Simplify: x² − 16x − 4
Explanation: (x − 4)(x + 4)x − 4 = x + 4, for x ≠ 4.
2. Simplify: 3x² + 6x3x
Explanation: 3x(x + 2)3x = x + 2, for x ≠ 0.
3. Simplify: x² + 5x + 6x + 2
Explanation: Factor the numerator: (x + 2)(x + 3)x + 2 = x + 3, for x ≠ −2.
4. Simplify: x² − 4x² + 5x + 6
Explanation: Factor both: (x − 2)(x + 2)(x + 2)(x + 3). Cancel the shared (x + 2) factor: x − 2x + 3, for x ≠ −2, −3.
5. For what value of x is the expression x + 5x − 3 undefined?
Explanation: A rational expression is undefined wherever its denominator equals 0: x − 3 = 0, so x = 3.
4. Recognizing Equivalent Expressions
A common SAT question style gives you an expression and asks which of four answer choices is equivalent to it. The most reliable approach is to fully simplify the given expression first, then match it to the answer choices, rather than trying to simplify each choice separately.
Worked Example: Which expression is equivalent to (x + 3)(x − 3) + 9?
(x + 3)(x − 3) + 9 = (x² − 9) + 9 = x²
1. Which expression is equivalent to 3(2x − 4) + 5x?
Explanation: Distribute: 6x − 12 + 5x. Combine like terms: 11x − 12.
2. Which expression is equivalent to (x + 3)(x − 3) + 9?
Explanation: This is the worked example above: (x² − 9) + 9 = x².
3. Which expression is equivalent to 2x(x + 4) − 2x²?
Explanation: Distribute: 2x² + 8x − 2x². The 2x² terms cancel, leaving 8x.
4. Which expression is equivalent to (x − 2)² + 4x?
Explanation: Expand: x² − 4x + 4 + 4x. The −4x and +4x terms cancel, leaving x² + 4.
5. Which of the following is NOT equivalent to 4x² − 12x + 9?
Explanation: 4x² − 12x + 9 factors as (2x − 3)². Since squaring removes the sign, (3 − 2x)² is also equal, both give the same expanded result. But (2x + 3)² expands to 4x² + 12x + 9, with a positive middle term, so it is NOT equivalent.
Common Mistakes to Avoid
- Forgetting the middle term when squaring a binomial. (a + b)² is not a² + b², it's a² + 2ab + b². Skipping the middle term is one of the most common algebra errors at every level.
- Canceling terms instead of factors in a rational expression. You can only cancel a factor that multiplies the entire numerator and entire denominator, never a term that's only added or subtracted within a larger expression.
- Forgetting to state the domain restriction on a simplified rational expression. Even after simplifying (x−3)(x+3)x+3 to x − 3, the original expression was still undefined at x = −3, and some questions expect you to note that restriction.
- Missing a negative sign when factoring out a GCF. If every term is negative, or the leading coefficient is negative, factor out a negative GCF carefully and double-check the signs of what remains inside the parentheses.
- Assuming an expression that "looks similar" is automatically equivalent. (2x − 3)² and (2x + 3)² look alike but expand to different expressions because of the sign inside. Always expand or factor completely before declaring two expressions equal.
Frequently Asked Questions
Do I need to memorize the special product patterns?
Yes, they save significant time. (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a − b)(a + b) = a² − b² show up constantly enough that recognizing them on sight, rather than re-deriving them with FOIL every time, is worth the memorization.
How can I check my work on a "which expression is equivalent" question?
Pick a simple number for x, like 2 or 3, and evaluate both the original expression and your simplified answer at that value. If they give the same result, your simplification is very likely correct. This works as a quick check even if you're not fully confident in your algebra.
Why does factoring matter if I can just expand everything instead?
Factoring reveals information that expanded form hides, such as a rational expression's undefined values, a quadratic's roots, or a shared factor that cancels between a numerator and denominator. Many Advanced Math questions are specifically designed so that factoring first is dramatically faster than working with the expanded form.
Where can I practice more problems like these?
The Equivalent Expressions quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.