Distribute, combine, and rewrite expressions

Two expressions are equivalent when they give the same value for every allowed value of their variables. For example,

The two sides look different, but distributing the $3$ turns the left side into the right side, so they give the same output for every $x$.

You'll recognize this kind of question from wording like:

• Which expression is equivalent to ...?
• The expression can be written in the form ...
• What is the coefficient of ... after rewriting?

None of these asks you to find a value of $x$. You're changing the form, or picking the matching form, and $x$ stays a variable the whole time. If the question gives you an equation and asks you to find an unknown, that's solving, and you balance both sides as in Solve linear equations.

Start with full-curve overlap when the question looks like the opening example:

• The expression has only one variable, like $x$.
• The answer choices are complete expressions, so each one can go straight into Desmos.
• Expanding by hand would be long, with several products or minus signs to track.

Here's how it goes. Type the original on line 1. Type one choice on line 2, then replace it with each of the other choices in turn. The right choice sits exactly on top of the original across the whole visible graph.

Why isn't one shared point enough? Two different expressions can give the same output at a single input. For example, $x^2$ and $2x$ both equal $4$ when $x=2$, but at $x=3$ they give $9$ and $6$. So a curve that only crosses the original proves nothing. You need the whole curve to match.

Start by hand in these cases instead:

• One short distribution settles it, like turning $2(x+7)$ into $2x+14$.
• More than one variable remains, like in $4(a-b)+a$.
• The question asks you to produce a coefficient or write the form yourself, like "What is the value of $c$?"
• The expression has a variable in a denominator, like $\frac{x^2-4}{x-2}$, where $x$ can't be $2$. A graph can't show a restriction like that exactly.

Related: For more practice comparing expressions by graph, see Equivalent expressions by graph overlap.

The distributive property says that a factor outside parentheses multiplies every term inside:

It works the same way with subtraction:

A minus sign in front of parentheses works like a factor of $-1$. Writing the $-1$ in makes it clear that every sign inside flips:

Once the parentheses are gone, combine like terms, terms with exactly the same variable part. $4x^2$ and $-x^2$ are like terms. $4x^2$ and $4x$ aren't, because the powers of $x$ are different.

Here are both steps together:

You won't always need the grouping line in the middle, but it lays out the three like-term groups so you can check each one.

A polynomial is a sum of terms whose exponents are whole numbers, like $4x^2-3x+8$. Adding, subtracting, and multiplying polynomials uses the same two moves over and over: distribute completely, then combine like terms.

Add or subtract

When you add, the parentheses can come straight off. Then combine the matching terms:

Subtraction is where signs go missing. The minus sign applies to the whole second polynomial, so every one of its terms flips:

Look at the last term. Subtracting $-11$ gives $+11$.

Multiply

When you multiply two polynomials, every term in one has to multiply every term in the other. Two binomials give four products:

Write out all four products before you combine anything. That's what keeps the middle term, here $8x-3x=5x$, from going missing. The same every-term rule works for bigger polynomials too.