Advanced Math / Equivalent Expressions
Equivalent Expressions
Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, and simplifying rational expressions while respecting restrictions on denominators.
Least you need to know
- Factoring and expanding are inverse ways to rewrite polynomial expressions.
- Like terms can be combined only when they have the same variable part.
- A rational expression can be simplified by canceling common factors, not by canceling terms.
Key notation
- $(a+b)^2$ — A square of a binomial, equal to $a^2+2ab+b^2$
- $x^m x^n$ — A product of powers with the same base, equal to $x^{m+n}$
Worked example
- Consider $x^2+8x+15$.
- Find two numbers that multiply to 15 and add to 8: 3 and 5.
- Therefore, $x^2+8x+15=(x+3)(x+5)$.
Common mistakes
- Do not cancel terms across addition or subtraction in rational expressions.
- The expression $(x+a)^2$ is $x^2+2ax+a^2$, not $x^2+a^2$.
- When factoring, check by multiplying the factors back out.
How to recognize it
- The problem asks which expression is equivalent.
- The expression may contain products of binomials, perfect squares, differences of squares, powers, or rational expressions.
- The problem may ask for a coefficient or constant term after rewriting.
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