Equivalent expressions
Advanced Math is about 35% of SAT Math (13–15 of 44 questions), and equivalent expressions are one of its core skills. You will see multiple-choice questions that ask “which expression is equivalent to…”, and some questions where an unknown constant must be found by matching coefficients.
Expanding: distribute every term
To expand a product, multiply each term of the first bracket by each term of the second, then collect like terms. (x + 4)(x − 2) = x² − 2x + 4x − 8 = x² + 2x − 8. When a minus sign stands in front of a bracket, it changes the sign of every term inside: −(x² − 5) = −x² + 5. Three products appear so often that they are worth knowing by heart.
| Product | Expanded form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| (a − b)(a + b) | a² − b² |
Factoring
Factoring is expanding backward. Step 1: pull out the greatest common factor: 6x² + 9x = 3x(2x + 3). Step 2: look for a difference of squares: 49x² − 16 = (7x − 4)(7x + 4). Step 3: for x² + bx + c, find two numbers that multiply to c and add to b. For x² − 5x − 24, the numbers −8 and 3 work, so it is (x − 8)(x + 3). When the x² has a coefficient, such as 2x² + 7x + 3, try the combinations and check by expanding.
Exponent rules
Learn these rules and be careful with the words “power of a power”. A negative exponent means a reciprocal, a fractional exponent means a root. The denominator of the fraction is the root: x^(1/2) = √x and x^(2/3) = (∛x)². Remember that the rules work for the same base only.
| Rule | Example |
|---|---|
| x^a · x^b = x^(a+b) | x³ · x⁴ = x⁷ |
| x^a ÷ x^b = x^(a−b) | x⁷ ÷ x² = x⁵ |
| (x^a)^b = x^(ab) | (x³)² = x⁶ |
| (xy)^a = x^a · y^a | (2x)³ = 8x³ |
| x^(−n) = 1 / x^n | x^(−2) = 1/x² |
| x^(m/n) = (ⁿ√x)^m | x^(3/2) = (√x)³ |
Rational expressions: cancel factors, not terms
You can cancel only factors that multiply the whole top and the whole bottom. So factor first. (x² − 9)/(x + 3) = (x − 3)(x + 3)/(x + 3) = x − 3. You may never “cancel” the x in (x + 3)/x, because x is only a term of the sum. The two expressions are equal for every x except x = −3, where the original fraction is not defined: its graph is the line y = x − 3 with a hole (a missing point) at (−3, −6).
Matching coefficients
Two polynomials are equivalent for all x only if the coefficients of x², of x, and the constants match. If (x + a)(x + 4) is equivalent to x² + 9x + 20, then a + 4 = 9 and 4a = 20, so a = 5. This is the standard way to find an unknown constant in an equivalent-expression question. You can also check with a number: put x = 1 into both sides.
- x² + 5x − 17
- x² + 5x − 7
- 3x² + 5x − 7
- x² + 11x − 7
- Expand the product: 2x² + 8x − 3x − 12 = 2x² + 5x − 12.
- Subtract the bracket, changing every sign: −x² + 5.
- Add: 2x² − x² + 5x − 12 + 5 = x² + 5x − 7.
- (x − 6)(x + 4)
- (x − 8)(x + 3)
- (x + 8)(x − 3)
- (x − 12)(x + 2)
- Find two numbers with product −24 and sum −5: −8 and 3.
- So x² − 5x − 24 = (x − 8)(x + 3).
- Check by expanding: x² + 3x − 8x − 24 = x² − 5x − 24 ✓
- 8x⁶
- 12x⁶
- 16x⁶
- 64x⁶
- Apply the exponent to each factor: 16^(3/4) · (x⁸)^(3/4).
- 16^(1/4) = 2, and 2³ = 8.
- (x⁸)^(3/4) = x^(8 × 3/4) = x⁶.
- So the expression is 8x⁶.
- Expand the left side: 2a·x² − ab·x + 6x − 3b.
- Match x²: 2a = 6, so a = 3.
- Match the constants: −3b = −12, so b = 4.
- Match x: c = −ab + 6 = −12 + 6 = −6.
- Sign errors with a minus before a bracket−(x² − 5) = −x² + 5. Change every sign inside.
- (a + b)² = a² + b²The middle term is missing: (a + b)² = a² + 2ab + b². Test with a = b = 1.
- Cancelling terms instead of factors(x + 3)/x is not 4. Only cancel what multiplies the whole top and the whole bottom, after factoring.
- Mixing up the rules for adding and multiplying exponentsx³ · x⁴ = x⁷ (add), but (x³)⁴ = x¹² (multiply), and x³ + x⁴ cannot be simplified.
- Fractional exponentsx^(3/4): the bottom is the root, the top is the power. 16^(3/4) = (⁴√16)³ = 2³ = 8.
- Forgetting to check by a quick testPut x = 2 (or another easy number) into the original and into your choice. If the values differ, the expressions are not equivalent.
To test whether two expressions are equivalent, type each one on its own line. If the graphs lie exactly on top of each other, the expressions are equivalent; if you see two different curves, they are not. You can do this for all four answer choices and keep the one that matches the original. To find a missing constant, type it as a letter such as k: Desmos makes a slider, and you move it until the curves coincide.
- Type y = (2x − 3)(x + 4) − (x² − 5)
- Type y = x² + 5x − 7
- The second curve covers the first: they are equivalent
- Type y = x² + 5x − 17 to see a curve that does not match
Use Desmos when the algebra is long and you can test choices quickly. Hand factoring is faster for simple expressions, and exponent rules are often quicker to apply directly.
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