Equivalent Expressions

You already expand products in your head

Compute $13 \times 12$ without paper. Most people do it like this: thirteen twelves is thirteen tens plus thirteen twos — $130 + 26 = 156$. You split one factor, multiplied each piece, and added. Write the split out and it looks suspiciously like algebra:

$13 \times 12 = (10 + 3)(10 + 2) = 100 + 20 + 30 + 6 = 156$

Nothing about the amount changed — $156$ never stopped being $156$. You just changed its costume: from a product of two factors to a sum of four pieces. This unit — the doorway to Advanced Math — is that one move, done with $x$ in the mix. Two expressions are called equivalent when they’re the same number in every costume: whatever value $x$ takes, both come out equal. Not “equal at the value you happened to test” — equal at every single one.

You own all the ingredients already. F8 taught the distributive property and like terms; F7 taught that $x \cdot x = x^2$ (exponents add); F2 taught the greatest common factor. Advanced Math starts by pointing those three tools at each other.

The rectangle that multiplies

Replace the $10$ with an $x$: what is $(x+3)(x+2)$? Draw it as a rectangle — width $x + 3$, height $x + 2$ — and split each side at the plus sign. The rectangle breaks into four cells, and the total area can be read two ways:

That’s the whole rule, and now it can be said in one line — for any two sums:

Every term of the first parenthesis multiplies every term of the second: four products, no exceptions. (You may have heard this called FOIL — first, outer, inner, last. Fine as a memory aid, but the rectangle is the actual rule: it survives even when a parenthesis has three terms and the acronym runs out of letters.)

Watch the rectangle assemble itself and become a sum:

Now build rectangles yourself — and drag $x$:

Predict before you touch: for $(x+5)(x-2)$, which cell is negative, and what middle term do the two $x$-cells combine into? Type it and check. Then drag the $x$ slider slowly and watch the two readings under the rectangle — width times height, and the sum of the four cells. They move together, always. That is what equivalent means: you cannot find an $x$ that splits them.

The square that fools almost everyone

Try the $(x+4)^2$ preset. The eye wants $(x+4)^2 = x^2 + 16$ — squaring the first thing, squaring the second thing. It feels right because multiplication really does distribute over addition, so why wouldn’t squaring? But squaring isn’t one hit — it’s a product of two parentheses, $(x+4)(x+4)$, and the rectangle shows two $4x$ cells the shortcut never counts:

$(x+4)^2 = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$

When a rule feels plausible, test a number: at $x = 1$, $(1+4)^2 = 25$, but $1 + 16 = 17$. One disagreement and the costume is fake. That test — feed both expressions the same $x$ and compare — deserves its own bench:

Load the $(x+3)^2$ vs $x^2 + 9$ preset. The table agrees at $x = 0$ — and the lab still says no, and shows you the input that snaps the illusion. Agreement at a few points proves nothing; a single disagreement is fatal. Then try to build your own pair of perfect twins.

Factoring: the same street, walked backwards

Expanding turns $2x(3x+5)$ into $6x^2 + 10x$. Factoring walks home: given $6x^2 + 10x$, what product was it? Open each term up — $6x^2 = 2 \cdot 3 \cdot x \cdot x$ and $10x = 2 \cdot 5 \cdot x$ — and look for what they share: a $2$ and one $x$. That shared content is F2’s greatest common factor, now with $x$‘s in it. Pull it out front and write what’s left of each term inside:

$6x^2 + 10x = 2x(3x + 5)$

Watch that hunt happen at the board — each term opened up, the shared part found and pulled out front:

Check it the honest way: distribute $2x$ back through and you land exactly where you started. Factored form isn’t a new fact about the expression — it’s the same expression, dressed for a different job. (What jobs? That’s M2’s story: factored form is what cracks quadratic equations open.) Factor completely: $x(6x + 10)$ pulled out an $x$ but left a $2$ hiding inside — the GCF wants both.

Prove one to yourself in the lab above: type $6x^2 + 10x$ against your factored answer and watch the columns agree — everywhere.

Adding and subtracting: like terms, now with degrees

Polynomials add the way F8’s expressions did — like terms combine, and now “like” means same power of $x$: the $x^2$‘s live together, the $x$‘s live together, the constants live together. Addition just drops the parentheses. Subtraction is where points die:

$(3x^2 - 2x + 5) - (x^2 + 4x - 3) = 3x^2 - 2x + 5 \; {\color{#8f4a24}- x^2 - 4x + 3}$

The minus is a $-1$ multiplying the whole parenthesis, so it flips every sign inside — not just the first one. It feels like “subtract the first thing” because that’s how the sentence reads aloud; the algebra disagrees, and a quick $x = 0$ check exposes it (the constants alone: $5 - (-3)$ is $8$, not $2$). From there it’s grouping: $2x^2 - 6x + 8$.

The one thing to remember

An expression’s costume can change; its value never does. Expanding (every term times every term) and factoring out the GCF are the same street walked in opposite directions, and the area model draws the map. When two expressions claim to be equivalent — or when your own answer might be wrong — feed both the same $x$: one disagreement settles it.

The two moves

Finding the GCF of the terms: the biggest number dividing every coefficient × the lowest power of $x$ any term carries. $12x^3 - 8x^2 \Rightarrow$ GCF $= 4x^2 \Rightarrow 4x^2(3x - 2)$.

Adding / subtracting polynomials: combine terms of the same degree. A minus in front of a parenthesis multiplies everything inside by $-1$ first.

The equivalence test: two expressions are equivalent only if they agree at every $x$. To prove a claim false, one disagreeing value is enough. (To prove it true, do the algebra — matching at a handful of points is not proof.)