Equivalent Algebraic Expressions

Two groups simplified the same expression at the boards and arrived at answers that looked nothing alike — and both survived every test the class threw at them. That standoff is the topic. Two expressions are equivalent when they take the same value for every input both are allowed. “Looks different” is not evidence.

Two tests, one warning

To show two expressions are equivalent, simplify both until they match — that is a proof. To show they are not, find one input where they disagree — that is also a proof. But matching at a few inputs proves nothing by itself: it builds confidence, not certainty. Are $\frac{2x^2-4x-6}{x+1}$ and $8x^2-2x(4x-1)-6$ equivalent? The second collapses to $2x-6$. The first factors:

$$\frac{2(x-3)(x+1)}{x+1}=2x-6,x\ne -1$$

Almost equivalent — but at $x=-1$ the first expression has no value at all. That fine print is the theme of this whole page.

Radicals without fear

Because $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$ for $a,b\ge 0$, a radical can be split at any square factor: $\sqrt{24}=\sqrt{4}\sqrt{6}=2\sqrt{6}$. The same law tidies products — expand like any binomials, then simplify each radical:

$$(2+\sqrt{6})(3-\sqrt{12})=6-2\sqrt{12}+3\sqrt{6}-\sqrt{72}=6-4\sqrt{3}+3\sqrt{6}-6\sqrt{2}$$

No like terms among $\sqrt{3}$, $\sqrt{6}$, and $\sqrt{2}$ — so that is the finished form, even though it looks unfinished.

Rational expressions and their fine print

Rational expressions add, subtract, multiply, and divide like numerical fractions — common denominators and all — with one extra duty: state the restrictions, the input values that make any denominator zero, and state them from the original expression, before anything cancels. A cancelled factor is a crime scene tidied up; the restriction is the record that it happened.

Check your understanding

Simplify $\frac{2x}{4x^2+6x}-\frac{3}{2x+3}$.

Factor the first denominator: $4x^2+6x=2x(2x+3)$, so the first fraction is $\frac{2x}{2x(2x+3)}=\frac{1}{2x+3}$ — recording $x\ne 0$ and $x\ne -\frac{3}{2}$ on the way past. Then $\frac{1}{2x+3}-\frac{3}{2x+3}=\frac{-2}{2x+3}$, with $x\ne 0,-\frac{3}{2}$.

A claim built for Always, Sometimes, Never: “two expressions that agree at $x=1$ are equivalent.” Sometimes — and the counter-examples you build to say so are exactly the thinking Mistakes Are Data celebrates. These skills surface all semester, whenever simplifying is the road through a problem.

Curriculum connection

A3.1

simplify polynomial expressions by adding, subtracting, and multiplying Sample problem: Write and simplify an expression for the volume of a cube with edge length $2x+1$.

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A3.2

verify, through investigation with and without technology, that $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$, $a\ge 0$, $b\ge 0$, and use this relationship to simplify radicals (e.g., $\sqrt{24}$) and radical expressions obtained by adding, subtracting, and multiplying [e.g., $(2+\sqrt{6})(3-\sqrt{12})$]

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A3.3

simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values Sample problem: Simplify $\frac{2x}{4x^2+6x}-\frac{3}{2x+3}$, and state the restrictions on the variable.

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A3.4

determine if two given algebraic expressions are equivalent (i.e., by simplifying; by substituting values) Sample problem: Determine if the expressions $\frac{2x^2-4x-6}{x+1}$ and $8x^2-2x(4x-1)-6$ are equivalent.

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