On the test
Advanced Math is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder, so these questions appear all through the module, with the hardest ones near the end.
13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.
How I approached it
A small story from Aime
I approach these questions like a question about money change. If I give you one 1,000 franc note, or two 500 franc coins, or ten 100 franc coins, it looks different, but it is the same money. Nothing changed in value, only the shape. Mixing and tidying also works like this: if you rearrange the things in your school bag, the same things are still inside. Equivalent expressions are the same thing written in different clothes. Two expressions are equivalent if they give the same answer for every x. When I see this on the SAT, I ask which tool turns the clothes of one into the clothes of another: expanding, factoring, or combining. Then I check by putting in a number.
This skill is mostly careful, patient algebra, and you can practise it anywhere with a pencil. Every rule you learn here also makes the other Advanced Math questions easier.
What is inside this part
Using the distributive rule and like terms to rewrite an expression as a simple sum.
adding and subtracting polynomials · multiplying brackets · sign changes with a minus in front of a bracket
Rewriting a sum as a product, using a common factor, a difference of squares, or a trinomial pattern.
common factor · difference of squares · trinomials with a leading number
Simplifying or combining fractions that have x in the top or bottom.
cancelling common factors · adding fractions with a common bottom · restrictions on x
Rewriting using the power rules, roots, or the vertex form of a quadratic.
product and power rules · fractional exponents as roots · completing the square
How to know a question is from this part
You get one expression and four rewritten versions, and the question asks which one is equivalent, or asks for the value of a constant in the rewritten form.
- The word 'equivalent' in the question: 'Which expression is equivalent to ...?'
- No equals sign to solve and no question about x.
- Options that look different but use the same pieces: brackets in one, expanded in another.
- A fraction with x and four fraction options.
- 'If the expression is written as (x + a)² + b, what is a + b?' or 'What is the value of c?'
The lesson
1. What does equivalent mean?
Two expressions are equivalent if they have the same value for every possible x. For example 2(x + 3) and 2x + 6 are equivalent. Put x = 5 in both: 2(8) = 16 and 10 + 6 = 16. Put x = 0: 6 and 6. They always match.
The SAT gives you one expression and four options, and asks you which option is equivalent. You are not solving for x. There is no equals sign to solve. You are only changing the form.
2. Tool 1: Combining like terms and the distributive rule
Like terms have the same letter part: 3x and 5x are like terms, but 3x and 3x² are not. Add the numbers in front: 3x + 5x = 8x.
The distributive rule says a(b + c) = ab + ac. The number outside multiplies every term inside. When there is a minus sign in front of a bracket, it changes the sign of every term inside: -(x² - 4x + 6) = -x² + 4x - 6.
Which expression is equivalent to (3x² + 5x - 2) - (x² - 4x + 6)?
Show the steps
- Remove the second bracket and flip every sign inside: 3x² + 5x - 2 - x² + 4x - 6.
- x² terms: 3x² - x² = 2x².
- x terms: 5x + 4x = 9x.
- Numbers: -2 - 6 = -8.
2x² + 9x - 8
3. Tool 2: Multiplying two brackets (expanding)
To multiply (x + 3)(x + 5), multiply every term in the first bracket by every term in the second. Some students call this FOIL: First, Outer, Inner, Last.
First: x · x = x². Outer: x · 5 = 5x. Inner: 3 · x = 3x. Last: 3 · 5 = 15. Add: x² + 5x + 3x + 15 = x² + 8x + 15.
A picture helps. Draw a rectangle with sides (x + 3) and (x + 5) and split it in four parts. The total area is the answer.
Expand (2x - 1)(x + 4).
Show the steps
- First: 2x · x = 2x².
- Outer: 2x · 4 = 8x.
- Inner: -1 · x = -x.
- Last: -1 · 4 = -4.
- Add: 2x² + 8x - x - 4 = 2x² + 7x - 4.
2x² + 7x - 4
4. Tool 3: Factoring
Factoring is expanding backwards. You turn a sum into a product. There are four patterns to know.
Pattern 1, common factor: 6x² + 9x = 3x(2x + 3).
Pattern 2, difference of squares: a² - b² = (a - b)(a + b). So x² - 49 = (x - 7)(x + 7), and 4x² - 9 = (2x - 3)(2x + 3).
Pattern 3, simple trinomial x² + bx + c: find two numbers that multiply to c and add to b. So x² - 5x - 14: numbers -7 and 2, giving (x - 7)(x + 2).
Pattern 4, trinomial with a number in front, like 6x² + x - 12: multiply a × c = -72, find two numbers that multiply to -72 and add to 1 (they are 9 and -8), split the middle: 6x² + 9x - 8x - 12, group: 3x(2x + 3) - 4(2x + 3) = (3x - 4)(2x + 3).
Always check by expanding your answer. It takes a few seconds.
Factor 6x² + x - 12.
Show the steps
- a × c = 6 × (-12) = -72.
- Numbers that multiply to -72 and add to +1: 9 and -8.
- Split: 6x² + 9x - 8x - 12.
- Group: 3x(2x + 3) - 4(2x + 3) = (3x - 4)(2x + 3).
(3x - 4)(2x + 3)
5. Tool 4: Fractions with x (rational expressions)
To simplify a fraction with x, factor the top and the bottom, then cancel factors that are the same on top and bottom. You can cancel only whole factors (things multiplied), never terms that are added.
For (x² - 9) ÷ (x² + x - 12): top = (x - 3)(x + 3), bottom = (x + 4)(x - 3). Cancel (x - 3). The answer is (x + 3) ÷ (x + 4).
To add fractions you need a common bottom, just as with number fractions. 1/(x - 1) + 2/(x + 1): the common bottom is (x - 1)(x + 1) = x² - 1. The top is 1(x + 1) + 2(x - 1) = 3x - 1. The answer is (3x - 1) ÷ (x² - 1).
Simplify (x² - 9) ÷ (x² + x - 12).
Show the steps
- Factor top: (x - 3)(x + 3).
- Factor bottom: (x + 4)(x - 3).
- Cancel (x - 3).
(x + 3) ÷ (x + 4)
6. Tool 5: Exponent rules and completing the square
Exponent rules: x^a · x^b = x^(a+b). (x^a)^b = x^(ab). (xy)^n = x^n y^n. x^(-n) = 1 ÷ x^n. A power of 1/2 means square root. For example (2x³y)² · 3xy² = 4x⁶y² · 3xy² = 12x⁷y⁴.
Completing the square rewrites x² + bx + c as (x + b/2)² + (c - (b/2)²). For x² + 8x - 3: half of 8 is 4, and 4² = 16, so x² + 8x - 3 = (x + 4)² - 16 - 3 = (x + 4)² - 19.
This is useful because the vertex form shows the vertex (-4, -19) directly.
7. The test-a-number safety check
When you are not sure, choose an easy number for x (like 2 or -1, avoid 0 and 1 if they make many options match) and compute the original expression. Then compute each option. The one that matches is correct.
To be safe, test two numbers. One number can make a wrong option accidentally match.
This works because equivalent expressions agree on every x. If an option disagrees even once, it is wrong. In Desmos, type the original and an option, and see if the graphs sit on top of each other.
8. Common traps
Trap 1: (x + 3)² is not x² + 9. It is x² + 6x + 9.
Trap 2: forgetting to change all signs when subtracting a bracket.
Trap 3: cancelling terms instead of factors. (x + 3) ÷ (x + 4) cannot be reduced to 3/4.
Trap 4: when factoring x² - 5x - 14, using signs in the wrong order: (x + 7)(x - 2) gives +5x, not -5x.
Trap 5: the options that look right but have the sign of the middle term wrong. Always check the middle term.
See it
The rectangle with sides x + 3 and x + 5 is split into four parts. Add the four areas to expand.
| Pattern | Example | Factored |
|---|---|---|
| Common factor | 6x² + 9x | 3x(2x + 3) |
| Difference of squares | 4x² - 9 | (2x - 3)(2x + 3) |
| Simple trinomial | x² - 5x - 14 | (x - 7)(x + 2) |
| Perfect square | x² + 6x + 9 | (x + 3)² |
- (x + 3)(x + 5)
- Multiply every term by every term
- Result: x² + 8x + 15
- Use when options are sums
- x² + 8x + 15
- Find the numbers that multiply to c, add to b
- Result: (x + 3)(x + 5)
- Use when options are products
- 1Look at the options: sums or products?
- 2Choose a tool: combine, expand, or factor
- 3Do the algebra slowly with brackets
- 4Check by testing x = 2 (or another easy number)
- 5Pick the only option that matches
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