The big idea in one picture: multiplying (x+2)(x+3) is finding the area of a rectangle. Cut it into four cells, find each cell's area, and add — that is the answer, x2+5x+6.
You already know how to add and subtract algebraic expressions from Lesson 7.3, and you just sharpened the power rules in 7.4. Now we put them to work on the operation that builds almost everything later in algebra: multiplication. The secret is that every kind of expression multiplication — a single term times a single term, a term times a long polynomial, even two polynomials together — is the same one move repeated: multiply, then collect. And there is a picture, the area model, that shows you exactly why.
By the end of this lesson you will be able to multiply a monomial by a monomial by handling the numbers and the letters separately, hand a factor out to every term with the distributive law, multiply two polynomials by pairing every term with every term, and draw any of these as the area of a rectangle so the rule is never something you have to memorize blind. We keep our usual colors throughout: letters are blue, numbers (coefficients) are amber, exponents are purple, and a sign that flips is flagged in red.
7.5.1 Monomial times monomial
A monomial is a single product of a number and some letters with whole-number exponents — things like 3x2 or −2x3. To multiply two of them, remember that multiplication can be reordered and regrouped however you like. So gather the numbers with the numbers, and the same-base letters with each other:
Two separate jobs are happening here. The coefficients are simply multiplied like ordinary numbers: 3 times −2 is −6. The letters combine by the product-of-powers rule from Lesson 7.4: when you multiply powers of the same base, you add the exponents, so x2 · x3 = x2+3 = x5. Think of it as merging two blocks: their sizes (the numbers) multiply, and their layers (the exponents) stack up.
Do the two jobs separately, then snap them together: (3x2)(−2x3) = −6x5.
Key idea
To multiply monomials: multiply the coefficients (numbers with numbers, signs included) and add the exponents of each shared letter (the product-of-powers rule). A letter that appears in only one factor just comes along for the ride.
Worked example — different letters present
Multiply (4a2b)(5ab3).
Multiply the coefficients: 4 · 5 = 20. numbers with numbers
Combine the a's: a2 · a1 = a3. 2 + 1 = 3
Combine the b's: b1 · b3 = b4. 1 + 3 = 4
Put it together: 20a3b4. each base handled on its own
Watch out — add exponents, don't multiply them
When you multiply two powers of the same base, the exponents add: x2 · x3 = x5, notx6. Multiplying the exponents is what you do when you raise a power to a power, (x2)3=x6 — a different rule from 7.4. And mind the signs: a positive times a negative is negative.
🎮 Try itMultiply two monomials
Set each coefficient and each exponent of x. Watch the numbers multiply and the exponents add to build the product. Try to recreate (3x2)(−2x3) = −6x5.
a · coeff3exp2
b · coeff-2exp3
7.5.2 Monomial times polynomial
What if the second factor has more than one term — a whole polynomial like 3x+4? Here the distributive law takes over. It says that an outside factor must be handed to every term inside the parentheses, one at a time:
2x(3x+4)
=
2x·3x
+
2x·4
=
6x2
+
8x
Notice each new term is just a monomial-times-monomial product — exactly the move you mastered in 7.5.1. The first piece, 2x·3x, has coefficients 2·3=6 and x1·x1=x2, giving 6x2. The second piece, 2x·4, is just 8x. The distributive law is simply the instruction that you must reach every term, missing none.
The strip of height 2x and width 3x+4 splits into two pieces. Their areas, 6x2 and 8x, are the two terms of the answer.
Key idea — the distributive law
For any quantities, a(b+c) = ab + ac: the outside factor multiplies each inside term. With more terms, reach all of them: a(b+c+d) = ab+ac+ad.
Worked example — a subtraction and a sign
Expand −3x(2x2−5).
Hand −3x to the first term: −3x · 2x2 = −6x3. −3·2=−6, x·x²=x³
Hand it to the second term: −3x · (−5) = +15x. negative times negative is positive
Combine: −6x3 + 15x. no like terms to merge
Watch out — reach every term, and carry the sign
The most common slip is forgetting the last term, especially a lone constant: 2x(3x+4) is 6x2+8x, not just 6x2. The second slip is dropping a minus sign — a negative outside factor flips the sign of every term it touches.
🎮 Try itDistribute a monomial
Set the outside factor mx and the two inside coefficients. Watch the arrows hand the factor to each term, and the area strip split into its two pieces. Recreate 2x(3x+4)=6x2+8x.
outside mx2
inside px3
+ q4
7.5.3 Polynomial times polynomial
Now both factors have several terms. The rule grows naturally out of the distributive law: every term of the first factor multiplies every term of the second, and then you add up all the products. For two two-term factors that is four products:
(a+b)(c+d)
=
ac
+
ad
+
bc
+
bd
Why four? Because a reaches both c and d, and so does b — two terms times two terms makes 2 × 2 = 4 products. Once you have all of them, the final step is to combine like terms, exactly as in Lesson 7.3. Watch it happen with real numbers:
The two middle products, 3x and 2x, are like terms — both are a number of x's — so they merge into 5x. The x2 term and the constant 6 have no partners, so they stay as they are. Many people remember the four products by the nickname FOIL — First, Outer, Inner, Last — but FOIL is just a memory aid for "every term times every term," and it only covers the two-by-two case. The real rule never changes: multiply all the pairs, then collect.
Key idea
To multiply two polynomials, multiply each term of one by each term of the other, keep every product (with its sign), then combine like terms. A factor with m terms times one with n terms gives m × n products before collecting.
Worked example — a subtraction inside
Expand (x+5)(x−2).
First:x·x = x2. x¹·x¹=x²
Outer:x·(−2) = −2x. keep the minus
Inner:5·x = 5x.
Last:5·(−2) = −10. number times number
Collect like terms −2x and 5x → 3x. Answer: x2+3x−10.
Watch out — combine like terms only at the end
Don't stop at x2+3x+2x+6 — that's the expansion, not the simplified answer. And only the genuine like terms merge: 3x and 2x combine, but x2 and x never do, because x² and x are different powers.
🎮 Try itEvery term times every term
Choose a product of two binomials (x+p)(x+q). The four products are listed, the two like terms are highlighted, and they combine into the final trinomial.
p2
q3
7.5.4 The area model
Everything above becomes obvious once you see it as area. The area of a rectangle is length times width. So a product like (x+2)(x+3) can be drawn as a rectangle whose width is x+2 and whose height is x+3. Cut the width at the x mark and the height at the x mark, and the rectangle falls into four smaller cells — one for each pair of pieces.
Each cell's area is one of the four products, and the total area is their sum — which is exactly the expanded expression. The picture makes "every term times every term" something you can literally point at: there is one cell for x·x, one for x·3, one for 2·x, and one for 2·3.
x
2
x
x²
2x
3
3x
6
The big rectangle, area (x+2)(x+3), is the sum of its four cells: x2 + 2x + 3x + 6 = x2 + 5x + 6.
The same picture explains the earlier sections too. A monomial times a polynomial is a rectangle with just one row of cells (one factor has a single term). A monomial times a monomial is a single cell — one rectangle, no cuts. The area model is the one idea underneath all of expression multiplication: break the rectangle into pieces, find each piece, add them up.
Key idea — area is the master plan
Draw the product as a rectangle; cut each side at its plus signs; the cells are the term-by-term products; the total area is the expanded expression. Every product rule in this lesson is just this picture, read off.
🎮 Try it ★The area model for (x + p)(x + q)
Slide the two constants. The rectangle re-cuts into four cells, each labelled with its area. Watch the two middle cells add up as like terms, and the total area become the trinomial x2+(p+q)x+pq.
top edge: x +2
side edge: x +3
★ The big ideas, in one breath
Every expression product is the same move: multiply, then collect. A monomial times a monomial — multiply the coefficients, add the exponents of each shared letter: (3x2)(−2x3)=−6x5. A monomial times a polynomial — the distributive law hands the outside factor to every inside term: 2x(3x+4)=6x2+8x. A polynomial times a polynomial — every term times every term, then combine like terms: (x+2)(x+3)=x2+5x+6. And all three are just the area of a rectangle cut into cells.
Coming up next — 7.6
Some products come up so often that they deserve shortcuts. In Lesson 7.6 you'll meet the multiplication formulas — patterns like (a+b)2 and (a+b)(a−b) — that let you skip the four-product work once you recognize the shape.
✎ Practice 7.5
Work each one out first, then open the answer to check your thinking.
Use the area model to expand (x+1)(x+7). Name the four cell areas.
Show answer
Cells: x2, 7x, 1x, 7. Total: x2+7x+x+7 = x2+8x+7.
Spot the error. A student writes (x+3)(x+2)=x2+6. What did they forget, and what is the correct answer?
Show answer
They only multiplied the firsts (x·x) and the lasts (3·2) and skipped the two middle products. The full work: x2+2x+3x+6 = x2+5x+6. Every term must meet every term.
🎯 Check yourself
Six questions to lock it in. Tap the answer you think is right.