ORIGINAL WORKED EXAMPLE
Subtract a polynomial
Simplify (4x² − 3x + 8) − (x² + 5x − 2).
- Distribute the subtraction: 4x² − 3x + 8 − x² − 5x + 2.
- Group matching powers: (4 − 1)x² + (−3 − 5)x + (8 + 2).
- Combine coefficients.
Answer 3x² − 8x + 10
SAT · Advanced Math · TOPIC LESSON
Equivalent expressions
Use algebraic structure to reveal the same quantity in a more useful form.
BY THE END, YOU CAN
- Expand and factor polynomials using structure.
- Simplify rational expressions without losing domain restrictions.
- Connect integer and rational exponents with radicals.
BEFORE YOU START
- Distribution, signed arithmetic and fractions.
- Meaning of a variable and a power.
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original SAT-style worked examples and practice
The teaching passages, worked examples and exercises in this lesson are original. They are not official College Board questions or past-exam reproductions. Blueprint and observed-practice analytics describe separate factual metadata; those source questions are not displayed here.
01 / UNDERSTAND & APPLY
Same values, with the same allowed inputs
Equivalent expressions have equal values for every input in their common stated domain. Their forms can reveal different features: expanded form exposes coefficients, factored form exposes zeros, and completed-square form exposes a quadratic minimum or maximum. An expression is not an equation to solve until it is set equal to something.
Domain restrictions matter. (x² − 9)/(x − 3) equals x + 3 only when x ≠ 3. The simplified formula can be evaluated at 3, but the original expression cannot; cancellation does not restore a missing input.
02 / UNDERSTAND & APPLY
Combine like terms and distribute completely
Like terms have the same variable powers: 3x² and −5x² combine, but 3x² and 5x do not. Addition and subtraction combine coefficients. Multiplication distributes every term in one factor over every term in the other. The sign outside parentheses applies to the entire group.
ORIGINAL WORKED EXAMPLE
Multiply two binomials
Expand (2x − 3)(x + 4).
- Distribute 2x over the second factor: 2x² + 8x.
- Distribute −3 over the second factor: −3x − 12.
- Combine the x terms: 8x − 3x = 5x.
Answer 2x² + 5x − 12
03 / UNDERSTAND & APPLY
First look for a common factor
Factoring reverses multiplication. Extract the greatest shared numerical factor and the lowest shared power of each variable. A negative common factor can make the remaining leading coefficient positive. Always expand your factors to check them.
The expression 6x³ − 15x² becomes 3x²(2x − 5). The factor x² accounts for a double zero if the polynomial is later set equal to zero; that is additional information the factored form reveals.
04 / UNDERSTAND & APPLY
Squares, trinomials and grouping
The difference of squares identity is a² − b² = (a − b)(a + b). A sum of squares does not factor by that identity over the reals. The perfect-square identities are a² ± 2ab + b² = (a ± b)². For x² + bx + c, seek numbers that add to b and multiply to c.
For a nonmonic quadratic, check both the product and the middle coefficient. Grouping can expose a shared binomial in higher-degree polynomials; not every factoring problem has degree two.
ORIGINAL WORKED EXAMPLE
A nonmonic quadratic
Factor 6x² + 7x − 3.
- The product of leading and constant coefficients is −18. The pair 9 and −2 multiplies to −18 and adds to 7.
- Split the middle: 6x² + 9x − 2x − 3.
- Group: 3x(2x + 3) − 1(2x + 3).
- Extract the shared binomial: (3x − 1)(2x + 3).
Answer (3x − 1)(2x + 3)
ORIGINAL WORKED EXAMPLE
A higher-degree shared structure
Factor x³ + 2x² − 9x − 18.
- Group terms: x²(x + 2) − 9(x + 2).
- Factor the shared binomial: (x + 2)(x² − 9).
- Apply difference of squares to the second factor.
Answer (x + 2)(x − 3)(x + 3)
05 / UNDERSTAND & APPLY
Cancel factors, never pieces of a sum
A rational expression is a quotient of polynomials. Factor numerator and denominator, record where the original denominator is zero, and cancel only shared nonzero factors. In (x + 4)/x, x is not a factor of the entire numerator, so it cannot be canceled from the sum.
To add rational expressions, use a common denominator and combine the entire numerators. For instance, 1/(x + 1) + 1/(x − 1) = 2x/(x² − 1), with x ≠ ±1. Multiplication instead combines numerators and denominators as products.
ORIGINAL WORKED EXAMPLE
Simplification with excluded inputs
Simplify (x² − 4x)/(x² − 16).
- Factor numerator as x(x − 4) and denominator as (x − 4)(x + 4).
- The original denominator excludes x = 4 and x = −4.
- Cancel the factor x − 4 only on the remaining domain.
Answer x/(x + 4), with x ≠ 4 and x ≠ −4
06 / UNDERSTAND & APPLY
Integer exponent laws have conditions
For a common base, multiplication adds exponents and division subtracts exponents: aᵐaⁿ = aᵐ⁺ⁿ and aᵐ/aⁿ = aᵐ⁻ⁿ when a ≠ 0 for division. A power of a power multiplies exponents. The zero power a⁰ equals 1 for a ≠ 0; a negative power a⁻ⁿ means 1/aⁿ, not a negative number.
A power distributes across multiplication, (ab)ⁿ = aⁿbⁿ, but not across addition. For example, (x + 2)² = x² + 4x + 4, not x² + 4. Parentheses distinguish (−3)² = 9 from −3² = −9.
07 / UNDERSTAND & APPLY
Radicals and rational powers
For positive a, a^(m/n) = (the nth root of a)^m, with integers m,n and n > 0. For real even roots, the radicand must be nonnegative; odd roots can accept negative inputs. The principal square root is nonnegative, so √(x²) = |x|, not always x.
Use positive-base assumptions when applying fractional exponent laws without further case analysis. A negative exponent still means a reciprocal. Reduce rational exponents and state restrictions instead of assuming every rule valid for arbitrary negative bases.
ORIGINAL WORKED EXAMPLE
A rational exponent exactly
Evaluate 32^(3/5).
- The fifth root of 32 is 2 because 2⁵ = 32.
- Cube that root: 2³ = 8.
- The base is positive, so either root-then-power or power-then-root is valid.
Answer 8
ORIGINAL WORKED EXAMPLE
A radical with a variable
Simplify √(18x²) for real x.
- Factor 18x² as 9 × 2 × x².
- The principal square roots of 9 and x² are 3 and |x|.
- Leave the nonsquare factor 2 under the root.
Answer 3|x|√2
08 / UNDERSTAND & APPLY
Rewrite to expose a feature
The expression x² − 6x + 13 can be written (x − 3)² + 4 because (x − 3)² = x² − 6x + 9. The second form displays a minimum of 4 over real x. Likewise 50(1.08)^(t/2) shows an 8% growth factor every two time units, while 50(√1.08)^t shows the one-unit factor. Equivalent forms answer different questions.
Numerically testing two expressions at one input can disprove equivalence if their values differ, but agreement at one or even several inputs is not an algebraic proof. Exact expansion or factoring establishes the identity over its domain.
09 / UNDERSTAND & APPLY
Original practice: preserve value and domain
State restrictions when canceling or using roots.
ORIGINAL PRACTICE · 1
Expand (x − 5)².
Reveal answer & reasoning
x² − 10x + 25
Multiply (x − 5)(x − 5); the two cross terms each contribute −5x.
ORIGINAL PRACTICE · 2
Factor 12x³ + 8x².
Reveal answer & reasoning
4x²(3x + 2)
Both terms contain 4x², and division leaves 3x and 2.
ORIGINAL PRACTICE · 3
Factor x² − 49.
Reveal answer & reasoning
(x − 7)(x + 7)
This is a difference of squares, x² − 7².
ORIGINAL PRACTICE · 4
Simplify (x² + 5x + 6)/(x + 2).
Reveal answer & reasoning
x + 3, for x ≠ −2
The numerator factors as (x + 2)(x + 3); the original denominator excludes −2.
ORIGINAL PRACTICE · 5
For a > 0, simplify a^(2/3)a^(1/3).
Reveal answer & reasoning
a
Add exponents for the same base: 2/3 + 1/3 = 1.
ORIGINAL PRACTICE · 6
Evaluate 16^(−1/2).
Reveal answer & reasoning
1/4
The negative exponent takes the reciprocal of √16 = 4.
ORIGINAL PRACTICE · 7
Is √(x²) = x for x = −6?
Reveal answer & reasoning
No; √(x²) = 6
The principal square root is nonnegative, giving |−6| = 6.
ORIGINAL PRACTICE · 8
Rewrite x² + 8x + 19 to reveal its minimum.
Reveal answer & reasoning
(x + 4)² + 3; minimum 3
Completing the square adds and subtracts 16; the squared term is at least zero.
ORIGINAL PRACTICE · 9
Simplify 3/(x − 2) + 1/(x − 2).
Reveal answer & reasoning
4/(x − 2), x ≠ 2
The denominators agree, so add numerators only. The denominator cannot be zero.
ORIGINAL PRACTICE · 10
Factor x³ − 4x² + x − 4.
Reveal answer & reasoning
(x − 4)(x² + 1)
Grouping gives x²(x − 4) + 1(x − 4). The real sum of squares x² + 1 does not factor by a difference-of-squares rule.
KEEP THE BIG IDEAS
Quick revision
- Equivalent forms reveal different properties of the same expression.
- Cancel complete factors and retain original excluded inputs.
- Fractional powers and square roots require attention to sign and domain.
READ WITH CONTEXT
References & scope
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
- College Board: SAT Math content domains ↗
Official skill taxonomy reference. All teaching prose, examples, diagrams and practice in this lesson are original; this is not a past-question collection.
Explore this topic in context
Equivalent expressionsSAT topic analytics ↗
SAT analytics distinguish official blueprint weights from independently classified linear-practice question occurrences. These are not historical administered-exam frequencies.