Dividing algebraic expressions

Division of powers. Just as $x^m \times x^n = x^{m+n}$ adds exponents, division subtracts them, provided the term being divided has an exponent at least as large as the divisor's:

where: $x$ is the common base (the repeated variable), $m$ is the exponent in the term being divided, and $n$ is the exponent in the divisor.

Monomial divided by monomial. Divide the numbers, then subtract exponents on each matching letter separately:

Coefficients divide the ordinary way; each variable's exponent is handled on its own — $x$'s exponents subtract independently of $y$'s.

Polynomial divided by monomial. Split the division across every term of the polynomial before simplifying anything. This is the distributive law read backwards:

where: $a$, $b$, $c$ are the terms of the polynomial and $m$ is the single monomial divisor. Once split, each small fraction is simplified with the exponent-subtraction rule above, term by term, keeping the original $+$ or $-$ sign attached to each piece. A boundary worth stating plainly: this chapter only divides by a single-term divisor. An expression like $(x^2+5x+6)\div(x+2)$, where the divisor itself has more than one term, needs a long-division-style process that belongs to later classes — you will not be asked for it here, and trying the term-by-term shortcut on a multi-term divisor gives a wrong answer. It also helps to notice that every division fact is a multiplication fact wearing a different hat. If $15x^4y^2 \div 5x^2y^2 = 3x^2$, then automatically $5x^2y^2 \times 3x^2 = 15x^4y^2$ — the two statements say the same thing about how the three quantities relate, just read in opposite directions. This is exactly why "multiply the quotient back by the divisor" works as a check: you are simply confirming the matching multiplication fact is true. Students who already trust their multiplication of monomials (coefficients multiply, exponents on matching letters add) find division falls into place quickly, because it is that same skill run backwards — subtracting instead of adding, and dividing instead of multiplying. One more habit worth building early: before dividing, glance at every term of the polynomial and confirm each variable's exponent is at least as large as the divisor's. If even one term fails that check, an exponent subtraction would go negative, which signals either a misread question or a term that genuinely falls outside what this chapter expects. Scanning first, rather than discovering the problem mid-calculation, saves time in a timed test.

Divide $8a^3b - 12a^2b^2 + 4ab$ by $4ab$.

Step 1. Split into one fraction per term:

Step 2. Simplify each piece. $8\div 4 = 2$, with $a^{3-1}=a^2$ and $b^{1-1}=1$, giving $2a^2$. Next, $12\div 4 = 3$, with $a^{2-1}=a$ and $b^{2-1}=b$, giving $3ab$. Finally $4ab\div 4ab = 1$ — a term divided by itself, not zero.

Step 3. Combine with the original signs: $2a^2 - 3ab + 1$.

Step 4 (check). Multiply the quotient back by $4ab$: $4ab(2a^2-3ab+1) = 8a^3b - 12a^2b^2 + 4ab$, which matches the original polynomial. Answer: $(8a^3b - 12a^2b^2 + 4ab) \div 4ab = 2a^2 - 3ab + 1$.

Competitive stretch

Divide $18x^5y^2 - 24x^4y^3 + 6x^3y^4$ by $6x^3y^2$. Split term by term: $18\div 6 = 3$ with $x^{5-3}=x^2$, $y^{2-2}=1$, giving $3x^2$; $24\div 6=4$ with $x^{4-3}=x$, $y^{3-2}=y$, giving $4xy$; $6\div 6=1$ with $x^{3-3}=1$, $y^{4-2}=y^2$, giving $y^2$. Combining signs gives $3x^2 - 4xy + y^2$. Check: $6x^3y^2(3x^2-4xy+y^2) = 18x^5y^2 - 24x^4y^3 + 6x^3y^4$ ✓. Answer: $3x^2 - 4xy + y^2$. Notice the pattern across both worked problems: the divisor's exponent on each letter is always subtracted, never the other way round, and the divisor's coefficient always divides into every term's coefficient exactly, because these problems are built so the division comes out clean. If a "quick mental check" ever gives a fraction or a negative exponent on a term that should divide evenly, that is a signal to re-read the question rather than to force an answer — a genuinely well-posed polynomial-by-monomial question at this level always divides every term without leftover fractions.

• Dividing only the first term of the polynomial and leaving the remaining terms unchanged, instead of dividing every term by the monomial.
• Subtracting exponents backwards — writing $x^4\div x^2$ as anything other than $x^{4-2}=x^2$; the divisor's exponent always comes off the term's exponent, never the reverse.
• Forgetting that a term divided by an identical term gives $1$, not $0$ — for example $4ab\div 4ab = 1$.
• Losing track of a minus sign on a middle term when splitting the polynomial into separate fractions.
• Trying to use the same term-by-term shortcut when the divisor itself has more than one term; that only works for a single monomial divisor.

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