Dividing One Polynomial by Another, Step by Step
Learn to multiply monomials and binomials, expand a binomial by a trinomial, divide monomials using index laws, and carry out polynomial long division with a remainder.
Why does dividing polynomials look exactly like long division?
Because it is long division — the same algorithm, with powers of in place of powers of ten.
When you divide by , you ask how many sixes fit the leading digit, subtract, bring down the next digit, and repeat. When you divide by , you ask what times gives , subtract, bring down the next term, and repeat.
The arithmetic version lines up hundreds, tens and units. The algebraic version lines up , and constants. The bookkeeping is identical, and once you notice that, the method stops being a new thing to memorise.
This page covers the second part of the ICSE Class 8 Mathematics chapter on algebraic expressions.
When you divide by , you ask how many sixes fit the leading digit, subtract, bring down the next digit, and repeat. When you divide by , you ask what times gives , subtract, bring down the next term, and repeat.
The arithmetic version lines up hundreds, tens and units. The algebraic version lines up , and constants. The bookkeeping is identical, and once you notice that, the method stops being a new thing to memorise.
This page covers the second part of the ICSE Class 8 Mathematics chapter on algebraic expressions.
Formula
What are the index laws for multiplying and dividing?
Multiplying and dividing algebraic terms rests on three rules of indices:
Multiply the coefficients, add the exponents of each variable. Divide the coefficients, subtract the exponents.
Worked example 1 — monomial times monomial.
Coefficients: . Powers of : . Powers of : .
Worked example 2 — monomial divided by monomial.
Coefficients: . Powers of : . Powers of : .
Worked example 3 — when a variable cancels completely.
Since , the disappears. Writing and then simplifying is more honest than silently dropping the , and it explains why is needed at all.
Worked example 4 — monomial times a bracket. This is the distributive law:
Every term inside the bracket is multiplied, including the last. Multiplying only the first term is the standard error here, and it is worth counting the terms in your answer against the terms in the bracket — three in, three out.
Worked example 5 — a bracket divided by a monomial.
Each term is divided separately. The last term gives , and writing nothing there instead of is a mistake that costs a whole term.
Multiply the coefficients, add the exponents of each variable. Divide the coefficients, subtract the exponents.
Worked example 1 — monomial times monomial.
Coefficients: . Powers of : . Powers of : .
Worked example 2 — monomial divided by monomial.
Coefficients: . Powers of : . Powers of : .
Worked example 3 — when a variable cancels completely.
Since , the disappears. Writing and then simplifying is more honest than silently dropping the , and it explains why is needed at all.
Worked example 4 — monomial times a bracket. This is the distributive law:
Every term inside the bracket is multiplied, including the last. Multiplying only the first term is the standard error here, and it is worth counting the terms in your answer against the terms in the bracket — three in, three out.
Worked example 5 — a bracket divided by a monomial.
Each term is divided separately. The last term gives , and writing nothing there instead of is a mistake that costs a whole term.
How do you multiply two binomials?
Every term in the first bracket multiplies every term in the second. Two terms times two terms gives four products, which then collect.
Worked example 1.
The four products:
-
-
-
-
The two middle terms collected to , which is a coefficient of and must still be written as a term.
**A check at .** The brackets give . The answer gives . They agree.
Worked example 2 — both signs negative.
The constant is because , while the middle term is negative. A positive constant with a negative middle term is the signature of two negative factors, and it is worth recognising for the factorising work that comes later.
Worked example 3 — binomial times trinomial.
Two terms times three terms gives six products:
- , ,
- , ,
Collecting:
Counting products before you start tells you how many to expect: two times three is six. If you have written five, one is missing.
Worked example 4 — three brackets.
Multiply two at a time. First , then
Checking at : the brackets give , and the answer gives .
Worked example 1.
The four products:
-
-
-
-
The two middle terms collected to , which is a coefficient of and must still be written as a term.
**A check at .** The brackets give . The answer gives . They agree.
Worked example 2 — both signs negative.
The constant is because , while the middle term is negative. A positive constant with a negative middle term is the signature of two negative factors, and it is worth recognising for the factorising work that comes later.
Worked example 3 — binomial times trinomial.
Two terms times three terms gives six products:
- , ,
- , ,
Collecting:
Counting products before you start tells you how many to expect: two times three is six. If you have written five, one is missing.
Worked example 4 — three brackets.
Multiply two at a time. First , then
Checking at : the brackets give , and the answer gives .
How do you carry out polynomial long division?
Arrange both polynomials in descending powers, then divide leading term by leading term, multiply back, subtract, and bring down.
Worked example 1 — an exact division. Divide by .
Step 1. . This is the first term of the quotient.
Step 2. . Subtract:
Step 3. Bring down , giving .
Step 4. , and . Subtract to get .
Check by multiplying back: . Correct.
Worked example 2 — with a remainder. Divide by .
Step 1. , and . Subtracting leaves .
Step 2. Bring down , giving .
Step 3. , and . Subtracting: .
The degree of is lower than the degree of , so division stops.
Check with the division identity:
Correct. A negative remainder is perfectly acceptable in algebra, unlike in whole-number division where remainders are kept non-negative.
Worked example 3 — a cubic. Divide by .
Step 1. , and . Subtracting: .
Step 2. Bring down , giving . Now , and . Subtracting: .
Step 3. Bring down , giving . Then , and . Subtracting gives .
So , and since , the cubic factorises completely as .
The two habits that prevent most errors. First, subtract, do not add — the sign of every term in the product must be changed, and writing the product in a bracket with a minus in front makes that visible. Second, leave a gap for missing powers. Dividing by is far easier written as , because the columns then line up and nothing is accidentally brought down into the wrong place.
When to stop. Division ends when the remainder's degree is lower than the divisor's. Stopping early leaves the answer incomplete; carrying on past that point produces fractions and is not what the question wants.
Worked example 1 — an exact division. Divide by .
Step 1. . This is the first term of the quotient.
Step 2. . Subtract:
Step 3. Bring down , giving .
Step 4. , and . Subtract to get .
Check by multiplying back: . Correct.
Worked example 2 — with a remainder. Divide by .
Step 1. , and . Subtracting leaves .
Step 2. Bring down , giving .
Step 3. , and . Subtracting: .
The degree of is lower than the degree of , so division stops.
Check with the division identity:
Correct. A negative remainder is perfectly acceptable in algebra, unlike in whole-number division where remainders are kept non-negative.
Worked example 3 — a cubic. Divide by .
Step 1. , and . Subtracting: .
Step 2. Bring down , giving . Now , and . Subtracting: .
Step 3. Bring down , giving . Then , and . Subtracting gives .
So , and since , the cubic factorises completely as .
The two habits that prevent most errors. First, subtract, do not add — the sign of every term in the product must be changed, and writing the product in a bracket with a minus in front makes that visible. Second, leave a gap for missing powers. Dividing by is far easier written as , because the columns then line up and nothing is accidentally brought down into the wrong place.
When to stop. Division ends when the remainder's degree is lower than the divisor's. Stopping early leaves the answer incomplete; carrying on past that point produces fractions and is not what the question wants.
Exam tip
Exam tip: count your products and subtract in brackets
Multiply coefficients, add exponents: . Divide coefficients, subtract exponents: . And , so a variable can vanish entirely.
Count the products before you expand. Two terms times three terms gives six products; if you have five, one is missing.
When dividing a bracket by a monomial, divide every term. A term like gives , not nothing.
In long division, arrange in descending powers and insert zero terms for missing powers — write , not , so the columns line up.
Write the subtraction in a bracket with a minus in front, then change every sign. Subtracting is where long division goes wrong.
Stop when the remainder's degree is below the divisor's. A negative remainder is fine in algebra.
Always verify with , or check numerically at .
Count the products before you expand. Two terms times three terms gives six products; if you have five, one is missing.
When dividing a bracket by a monomial, divide every term. A term like gives , not nothing.
In long division, arrange in descending powers and insert zero terms for missing powers — write , not , so the columns line up.
Write the subtraction in a bracket with a minus in front, then change every sign. Subtracting is where long division goes wrong.
Stop when the remainder's degree is below the divisor's. A negative remainder is fine in algebra.
Always verify with , or check numerically at .
Did you know
A zero remainder is a factor found
In the third worked example, dividing by left a remainder of zero — and that zero says something far stronger than the division worked out neatly.
A zero remainder means is a factor. Once you know that, the cubic has been reduced to a quadratic you can factorise by inspection, and the whole expression opens up as .
There is a shortcut hiding here. Substitute into the original cubic: . The expression vanishes at , and it does so precisely because divides it exactly — at that factor is zero, so the whole product is zero.
This works in reverse too. Testing gives , so is a factor as well, found in one line of arithmetic rather than a page of division.
That connection between a value that makes the expression zero and a factor of the expression is one of the most useful ideas in all of algebra, and long division is how you first meet it.
A zero remainder means is a factor. Once you know that, the cubic has been reduced to a quadratic you can factorise by inspection, and the whole expression opens up as .
There is a shortcut hiding here. Substitute into the original cubic: . The expression vanishes at , and it does so precisely because divides it exactly — at that factor is zero, so the whole product is zero.
This works in reverse too. Testing gives , so is a factor as well, found in one line of arithmetic rather than a page of division.
That connection between a value that makes the expression zero and a factor of the expression is one of the most useful ideas in all of algebra, and long division is how you first meet it.
Key takeaways
Multiplying and dividing polynomials: quick revision
- Index laws: , , and .
- Monomials: and . When powers match, .
- Monomial times bracket: — every term, including the last.
- Bracket divided by monomial: ; the last term is , not nothing.
- Two binomials give four products: . Check at : and .
- — a positive constant with a negative middle term signals two negative factors.
- Binomial times trinomial gives six products: .
- Three brackets, two at a time: . At , both sides give .
- Long division is ordinary long division with powers of instead of powers of ten. Divide leading terms, multiply back, subtract, bring down.
- gives quotient , remainder .
- gives quotient , remainder . Check: .
- gives exactly, so the cubic is .
- Insert zero terms for missing powers, subtract in brackets, and stop when the remainder's degree drops below the divisor's. A negative remainder is acceptable.
- Always check with .
- A zero remainder means a factor, and substituting that value makes the whole expression zero: confirms divides the cubic.
Work three long divisions with a missing power in the dividend — that is the case examiners favour, and the zero-term habit is the only thing standing between you and a misaligned column.
- Monomials: and . When powers match, .
- Monomial times bracket: — every term, including the last.
- Bracket divided by monomial: ; the last term is , not nothing.
- Two binomials give four products: . Check at : and .
- — a positive constant with a negative middle term signals two negative factors.
- Binomial times trinomial gives six products: .
- Three brackets, two at a time: . At , both sides give .
- Long division is ordinary long division with powers of instead of powers of ten. Divide leading terms, multiply back, subtract, bring down.
- gives quotient , remainder .
- gives quotient , remainder . Check: .
- gives exactly, so the cubic is .
- Insert zero terms for missing powers, subtract in brackets, and stop when the remainder's degree drops below the divisor's. A negative remainder is acceptable.
- Always check with .
- A zero remainder means a factor, and substituting that value makes the whole expression zero: confirms divides the cubic.
Work three long divisions with a missing power in the dividend — that is the case examiners favour, and the zero-term habit is the only thing standing between you and a misaligned column.