Rewrite polynomial, rational, radical and exponential expressions into equivalent forms. It is part of Advanced Math in the Math section. The book teaches it in Chapter 20, Equivalent Expressions, with 19 practice questions, and tests it 8 more times across the two full-length practice tests.
About Chapter 20
Two expressions are equivalent when they have the same value for every value of the variable for which both are defined. For example, 2(x + 3) and 2x + 6 are equivalent. Equivalent-expression questions belong to the Advanced Math domain. They ask you to expand, factor, simplify exponents and radicals, combine rational expressions, match coefficients and divide polynomials. You rarely solve for x; you rewrite.
What the question looks like
Expand or factor: Which expression is equivalent to (2x − 3)(x + 4)?
Exponents and radicals: For x > 0, which expression is equivalent to x^3/2?
Rational expressions: Which expression is equivalent to 1/x + 1/(x + 2)?
Matching coefficients: If (ax + 1)(x − 3) = 2x^2 + bx − 3 for all values of x, what is the value of b?
Division and remainders: Which expression is equivalent to (x^2 + 4x + 7)/(x + 1)?
How to recognize it
The stem says "equivalent," "can be rewritten in the form" or "for all values of x."
There is no equation to solve, or the equation is an identity: both sides must match for every x.
The answer choices are expressions that look different but might be the same.
A condition such as "for x > 0" or "x ≠ 3" appears. It keeps radicals and denominators defined; it is not a signal to solve.
Common traps
Dropping the middle term when squaring a binomial: (a − b)^2 always has a −2ab term.
Losing a sign when distributing a negative: Put the subtracted expression in parentheses and distribute the minus sign to every term.
Adding exponents when you should multiply, or the reverse: Same base multiplied: add. Power of a power: multiply.
Flipping a rational exponent: In x^m/n, the bottom number n is the root.
Cancelling terms instead of factors: Factor first; cancel only factors that multiply the entire numerator and denominator.
Adding fractions straight across: Rewrite with a common denominator before combining.
Using the wrong sign in the remainder theorem: A divisor of x + 3 means evaluate at x = −3.
Video lessons
Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.
Equivalent expressions with the graphing calculator (Raymond P. Hunter, 5:40, captions). Watch for: Graphing two expressions to see whether they match.
A fast check for equivalent expressions (Erika | Math + Engineering, 4:46). Watch for: Testing a value of x in both expressions as a backup to algebra.
Practice this skill
The full method, worked examples and 19 practice questions with every answer choice explained are in Chapter 20 of the book and in the study workspace, where an AI coach can talk you through each one.
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