Two expressions are equivalent when they give the same value for every allowed x. On the SAT you rewrite expressions by distributing, combining like terms, factoring, using exponent rules and adding fractions. Many questions show one expression and ask which choice equals it. When an equation is true "for all x", the matching terms on each side must have equal coefficients.
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The key idea
Rewriting never changes the value, only the look. A handful of patterns cover most SAT questions:
(a+b)2=a2+2ab+b2a2−b2=(a−b)(a+b)
xm⋅xn=xm+n(xm)n=xmnxnm=nxm
If two polynomials are equal for all x, they are the same polynomial. So the x2 coefficients match, the x coefficients match, and the constants match. That is how you find unknown constants.
Worked examples
Example 1: expand and combine
Problem Which expression is equivalent to (2x−3)(x+4)−x(x−1)?
Expand the first product. Every term in the first group multiplies every term in the second.
(2x−3)(x+4)=2x2+8x−3x−12
Expand the second product, keeping the minus sign in front.
−x(x−1)=−x2+x
Combine like terms.
2x2+5x−12−x2+x=x2+6x−12
Answerx2+6x−12
Example 2: difference of squares
Problem Factor 9x2−25.
Both terms are perfect squares: 9x2=(3x)2 and 25=52.
9x2−25=(3x)2−52
Use a2−b2=(a−b)(a+b).
(3x)2−52=(3x−5)(3x+5)
Answer(3x−5)(3x+5)
Example 3: fractional exponents
Problem Simplify (16x8)43, where x>0.
Apply the power to each factor.
(16x8)43=1643⋅(x8)43
1643 means the fourth root of 16, cubed: 23=8.
1643=8
Multiply the exponents for the x part: 8⋅43=6.
(x8)43=x6
Answer8x6
Example 4 (SAT-hard): match coefficients
Problem If (ax+3)(2x−b)=4x2−4x−15 for all values of x, where a and b are constants, what is a+b?
Expand the left side.
(ax+3)(2x−b)=2ax2+(6−ab)x−3b
Match the x2 coefficients: 2a=4.
a=2
Match the constants: −3b=−15.
b=5
Check the middle term: 6−ab=6−10=−4. It matches, so the values are right.
Answera+b=7
Common mistakes
Squaring term by term.(x+5)2 is x2+10x+25, not x2+25. Fix: write it as (x+5)(x+5) and multiply every pair.
Losing the minus sign in front of a group.−x(x−1) is −x2+x. Fix: distribute the negative to every term inside.
Adding fractions by adding tops and bottoms.x2+x+13 is not 2x+15. Fix: get a common denominator first.
Canceling terms instead of factors. In x+3x2−9 you cannot cancel the x's. Fix: factor the top to (x−3)(x+3), then cancel the whole factor x+3.
Adding exponents when you should multiply.(x8)43 is x6, not x8.75. Fix: a power of a power multiplies; a product of powers adds.
Quick methods
Practice
5 SAT-style questions
Which expression is equivalent to 3(2x−5)−4(x−2)?
2x−7
2x−13
2x−23
10x−7
Show answer
Answer: 2x−7
6x−15−4x+8=2x−7. 2x−23 forgets that −4×−2=+8. 2x−13 multiplies only the x term by −4.
Which expression is equivalent to x2−10x+25?
(x−5)2
(x+5)2
(x−5)(x+5)
(x−25)(x−1)
Show answer
Answer: (x−5)2
(x−5)2=x2−10x+25. (x+5)2 has +10x, (x−5)(x+5)=x2−25, and (x−25)(x−1) has −26x.
Which expression is equivalent to x+3x2−9 for x=−3?
x−3
x+3
x−9
x2−3
Show answer
Answer: x−3
Factor the top: x+3(x−3)(x+3)=x−3. x2−3 comes from canceling terms instead of factors.
Which expression is equivalent to x2+x+13?
2x+15
x(x+1)5x+2
x(x+1)5x+3
x(x+1)5
Show answer
Answer: x(x+1)5x+2
Common denominator x(x+1): x(x+1)2(x+1)+3x=x(x+1)5x+2. 2x+15 adds tops and bottoms, which is never allowed.
Student-produced response: if 4x2+bx+9=(2x−3)2 for all values of x, what is the value of b?
Show answer
Answer: -12
Expand: (2x−3)2=4x2−12x+9. Match the x terms: b=−12. The middle term is 2⋅2x⋅(−3), which people often forget to double.
Frequently asked questions
How do I know if two expressions are equivalent?
Simplify both to the same form, or plug in a couple of test values. If they ever give different outputs for an allowed x, they are not equivalent. If they match after full simplification, they are. Graphing both in Desmos shows the same curve when they are equivalent.
What does "for all values of x" mean in a question?
It means the two sides are the same expression, not an equation to solve. That lets you match coefficients: the x2 terms are equal, the x terms are equal, and the constants are equal. Each match gives you an equation for an unknown constant.
Do I need to know fractional exponents for the SAT?
Yes. College Board's Advanced Math domain includes radical and exponential expressions. Know that x21=x and xnm is the nth root of xm, and be ready to rewrite between the two.
