Running powers and multiplication in reverse: dividing powers, zero and negative exponents, and a first look at factoring.
Point 4 of 5 in this lesson: 7.7.4 Polynomial divided by monomial
Every operation you have met comes in a matched pair: adding and subtracting, then multiplying and dividing. In the last few lessons you learned to build expressions — you multiplied powers, multiplied monomials, and even used the multiplication formulas to expand things in one quick step. This lesson runs the film backward. When you divide, you take an expression apart instead of putting it together, and almost everything you do is a rule you already know, simply turned around.
By the end you will be able to do five things: divide two powers with the same base by subtracting their exponents, make sense of the strange-looking a0 and a−n that fall out when you keep subtracting, divide a monomial by a monomial, divide a polynomial by a monomial term by term, and take your first look at factoring — rewriting a sum as a product. We keep one steady habit of color throughout: a variable (a letter) is blue, a coefficient (a number multiplier) is amber, an exponent is purple, and a warning is red.
A power is just a stack of equal factors written in shorthand: a5 means a·a·a·a·a — five copies of the same number multiplied together. So when you divide one power by another power of the same base, you are really writing one stack of a’s over another stack of a’s, and a fraction like that begs to be reduced.
Watch what happens with a5 ÷ a2. Write it as a fraction with the factors spelled out:
The bottom a’s simply erase the same number of top a’s. So you never have to spell out the stacks at all — you just subtract the exponents. The base stays exactly the same; only the count of factors changes.
To divide two powers with the same base, keep the base and subtract the exponents:
am ÷ an = am−n
This is the mirror image of the multiplication rule from Lesson 7.4, where multiplying powers added the exponents. Multiply → add; divide → subtract.
Simplify x7 ÷ x3.
The exponents subtract; they are not divided and the base is not divided. a6 ÷ a2 is a4 (because 6 − 2 = 4), not a3 and not just a. And the rule needs the same base: a5 ÷ b2 cannot be combined, because no b cancels an a.
Set the top exponent m and the bottom exponent n (with m at least n). Watch the bottom a’s cancel the top a’s, and read off how many are left.
The subtract-the-exponents rule is so neat that it is worth asking a daring question: what if the bottom exponent is just as big as the top one, or even bigger? The rule does not flinch — it keeps giving answers — and those answers force us to decide what a zero exponent and a negative exponent must mean.
Start with equal exponents. Anything (except zero) divided by itself is 1, so a3 ÷ a3 = 1. But the rule says subtract: 3 − 3 = 0, which gives a0. For the two ways to agree, we are pushed into a single conclusion: a0 = 1.
For any base a that is not 0, a0 = 1. It is not a special new fact we declared by hand — it is the only value that keeps the division rule honest, since an ÷ an is both a0 and plainly 1.
Now keep going below zero. Take a2 ÷ a5. Spelled out, that is two a’s over five a’s; the two on top cancel two on the bottom and leave 3 a’s stuck in the denominator, so the value is 1a3. The rule, meanwhile, says 2 − 5 = −3, giving a−3. Once again both must agree, so a negative exponent must mean a reciprocal.
A negative exponent flips the power into the denominator: a−n = 1an (for a not 0). The minus sign on the exponent does not make the number negative — it tells you to take the reciprocal.
The cleanest way to feel all of this at once is to walk down a staircase, dividing by the base at every step. Each step down lowers the exponent by exactly 1, and the values glide straight through 1 and on into fractions without any jump:
2−2 is 14, a small positive number — it is not −4 and not −¼. The minus sign rides on the exponent, where it means "reciprocal," not on the value itself.
Step down the staircase. Start high and divide by the base each step. Watch the exponent fall past 0 while the value glides smoothly into fractions — 20 lands on 1, not by decree but because every step just halves the one before.
A monomial is a single chunk: a number multiplied by some letters with exponents, like 6x5. Dividing one monomial by another is nothing new — it is just two jobs done side by side. The numbers (the coefficients) divide like ordinary numbers, and each matching letter follows the quotient rule you just learned: subtract the exponents.
Take 6x5 ÷ 2x2. Sort it into two separate questions: a number question and a letter question.
This is exactly the reverse of multiplying monomials, where you multiplied the coefficients and added the exponents. Going forward you can check any division by multiplying back: since 2x2 × 3x3 = 6x5, the division 6x5 ÷ 2x2 = 3x3 must be right.
Simplify 15y6 ÷ 5y2.
Pick coefficients that divide evenly and exponents with the top at least the bottom. See the two jobs done separately — numbers divide, exponents subtract — then recombined.
A polynomial is a sum of monomials — several chunks added together, like 6x3 + 4x2. To divide a whole polynomial by a single monomial, you use the same fairness rule that division has always followed: share the divisor with every term. Each term gets divided separately, and then you add the results back up.
Why is that allowed? Because a fraction with a sum on top splits cleanly: A + BC = AC + BC. It is the distributive law you used to expand products, now working for you on the way back.
Simplify (10a4 − 15a2) ÷ 5a.
The most common slip is dividing only the first term and copying the rest. Every term must be divided. (6x3+4x2) ÷ 2x is 3x2+2x, not 3x2+4x2. Also keep each term’s sign: a subtraction stays a subtraction.
Choose two top coefficients and the divisor 2x. Watch the division get handed to each term, then watch the two simpler answers come back together.
Step back and look at what division has shown you. When you divided (6x2 + 8x) ÷ 2x you got 3x + 4. Read that the other way and it says something striking: 2x times (3x + 4) gives back 6x2 + 8x. Division has quietly handed you a way to rewrite a sum as a product.
That is the whole idea of factoring. Multiplying (or expanding) takes a product and spreads it out into a sum: 2x(3x+4) = 6x2+8x. Factoring runs that same arrow backward: it looks at the sum 6x2+8x, finds what every term has in common, and pulls that shared factor out front to write the sum as a product again.
Finding what to pull out is pure division. Each term holds a copy of 2x: 6x2 = 2x · 3x and 8x = 2x · 4. The shared piece 2x is the greatest common factor; what is left after you divide each term by it goes inside the parentheses.
Factor 6x2 + 8x.
You have just taken your first factoring step: pulling out a common factor. Factoring is the engine behind simplifying fractions of expressions and solving equations, and there is much more to it — grouping, special patterns, and trinomials. The reverse trip you glimpsed here is the road ahead.
Flip the toggle to run the bridge either way: expand the product into a sum, or factor the sum back into a product by pulling out the common 2x.
Division is multiplication run in reverse. Two powers with the same base divide by subtracting exponents: am ÷ an = am−n. Keep subtracting and the pattern itself forces a0 = 1 and a−n = 1an. A monomial ÷ monomial splits into a number job and a letter job: divide the coefficients, subtract the exponents. A polynomial ÷ monomial shares the divisor with every term. And running all of this backward — writing a sum as a product by pulling out a common factor — is your first taste of factoring.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in, aimed at the traps people fall into. Tap the answer you think is right.
This lesson closes Stage 7 by running multiplication in reverse. It builds straight on the power rules from 7.4 and the monomial and polynomial multiplication of 7.5. Previous lesson: 7.6 “Multiplication Formulas: Shortcuts for Faster Work.” The first taste of factoring you met here is the doorway to simplifying rational expressions in Stage 9.