Every Term Must Meet Every Other Term
Learn to multiply monomials and polynomials without missing a product, divide to recover one side of a rectangle from its area, clear algebraic fractions with the LCM, and remove nested brackets in order.
How many products do you get when two binomials are multiplied?
Four — each of the two terms in the first bracket must multiply each of the two in the second. Producing only two or three of them is the single most common algebra error in this chapter.
So is four multiplications, not two. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: multiplying algebraic expressions, dividing them, simplifying algebraic fractions, and removing nested brackets.
So is four multiplications, not two. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: multiplying algebraic expressions, dividing them, simplifying algebraic fractions, and removing nested brackets.
How do you multiply algebraic expressions?
Multiply the coefficients, then add the indices of matching variables.
Monomial by monomial:
Polynomial by monomial — multiply the monomial into every term:
Polynomial by polynomial — multiply every term of the first by every term of the second, then collect like terms:
The four products were , , and , and the two middle ones were like terms that combined.
Another, with a subtraction:
The sign travels with each term, so gives and gives .
The rule that fails a lot of answers is the one for indices: add them, do not multiply them. , not — and is certainly not either, since the coefficients are both 1.
Monomial by monomial:
Polynomial by monomial — multiply the monomial into every term:
Polynomial by polynomial — multiply every term of the first by every term of the second, then collect like terms:
The four products were , , and , and the two middle ones were like terms that combined.
Another, with a subtraction:
The sign travels with each term, so gives and gives .
The rule that fails a lot of answers is the one for indices: add them, do not multiply them. , not — and is certainly not either, since the coefficients are both 1.
How do you divide algebraic expressions?
Divide the coefficients and subtract the indices.
Monomial by monomial:
Polynomial by monomial — divide every term separately:
Notice that the first result, , is the bracket we multiplied by in the previous section — division undoes multiplication, which is a useful way to check either one.
Application to a rectangle. If a rectangle's area is and one side is , the other side is
And if the area of a rectangle is with one side , the other side is .
The trap is dropping a term. Every term of the polynomial must be divided, so gives three terms in the answer — and the last one is , not , since .
Monomial by monomial:
Polynomial by monomial — divide every term separately:
Notice that the first result, , is the bracket we multiplied by in the previous section — division undoes multiplication, which is a useful way to check either one.
Application to a rectangle. If a rectangle's area is and one side is , the other side is
And if the area of a rectangle is with one side , the other side is .
The trap is dropping a term. Every term of the polynomial must be divided, so gives three terms in the answer — and the last one is , not , since .
How do you simplify an algebraic expression with fractions?
Take the LCM of the denominators, rewrite each fraction with it, then combine the numerators.
Worked example. Simplify . The LCM of 2 and 3 is 6:
Worked example. Simplify . The LCM is 12:
Worked example with brackets in the numerators. Simplify . The LCM is 10:
Worked example with a subtraction of a bracket. Simplify , with LCM 12:
The danger sits in that last example. When a fraction with a minus in front has a bracket on top, the minus must reach every term inside — so gives , not . Writing the bracket out before combining is what keeps it right.
Worked example. Simplify . The LCM of 2 and 3 is 6:
Worked example. Simplify . The LCM is 12:
Worked example with brackets in the numerators. Simplify . The LCM is 10:
Worked example with a subtraction of a bracket. Simplify , with LCM 12:
The danger sits in that last example. When a fraction with a minus in front has a bracket on top, the minus must reach every term inside — so gives , not . Writing the bracket out before combining is what keeps it right.
In what order do you remove nested brackets?
Innermost outwards, in the fixed order vinculum (bar), then ( ), then { }, then [ ].
Worked example.
Clear the vinculum first, treating as a single quantity:
The round bracket next, distributing the minus sign:
Then the curly bracket:
Finally the square bracket:
A second example:
A minus sign before a bracket changes every sign inside it — that is the whole difficulty, and it appears at each of the four stages. Rewriting the full expression after every step, as above, is what makes the sign changes visible rather than guessed at.
Worked example.
Clear the vinculum first, treating as a single quantity:
The round bracket next, distributing the minus sign:
Then the curly bracket:
Finally the square bracket:
A second example:
A minus sign before a bracket changes every sign inside it — that is the whole difficulty, and it appears at each of the four stages. Rewriting the full expression after every step, as above, is what makes the sign changes visible rather than guessed at.
Exam tip
Exam tip: counting your products before collecting terms
Algebraic multiplication and bracket removal are marked line by line, so keep the lines.
When multiplying two brackets, count the products you should get — two terms by two terms gives four, two by three gives six — and check you have written that many before collecting like terms.
Add indices when multiplying and subtract them when dividing. Never multiply the indices.
Dividing a polynomial gives as many terms as it started with, so check none has vanished.
For fractions, write the LCM and the multiplied numerators on one line, keeping brackets intact, then expand.
And remove brackets in the order bar, ( ), { }, [ ], rewriting the whole expression each time. When a minus precedes a bracket, change every sign inside it.
When multiplying two brackets, count the products you should get — two terms by two terms gives four, two by three gives six — and check you have written that many before collecting like terms.
Add indices when multiplying and subtract them when dividing. Never multiply the indices.
Dividing a polynomial gives as many terms as it started with, so check none has vanished.
For fractions, write the LCM and the multiplied numerators on one line, keeping brackets intact, then expand.
And remove brackets in the order bar, ( ), { }, [ ], rewriting the whole expression each time. When a minus precedes a bracket, change every sign inside it.
Did you know
Why does dividing by a monomial give back the bracket you multiplied?
Because multiplication and division are exact opposites, and the distributive property works in both directions.
Multiplying into spread it across both terms to give . Dividing that result by takes the same step backwards, term by term, and returns .
That makes every multiplication its own check. Multiply out, then divide the answer by one of your factors — if you do not get the other factor back, a product was missed somewhere.
Multiplying into spread it across both terms to give . Dividing that result by takes the same step backwards, term by term, and returns .
That makes every multiplication its own check. Multiply out, then divide the answer by one of your factors — if you do not get the other factor back, a product was missed somewhere.
Key takeaways
Multiplying and dividing expressions: quick revision
- Multiply coefficients and add indices: ; divide coefficients and subtract indices: .
- A monomial must be multiplied into every term of a polynomial, and every term must be divided too.
- Two binomials give four products: after collecting like terms.
- One side of a rectangle comes from dividing the area by the other side: .
- For algebraic fractions, use the LCM of the denominators and keep numerator brackets intact — a minus before a bracket reaches every term, so .
- Remove brackets in the order vinculum, ( ), { }, [ ], rewriting the expression at each step.
You will remember all of this far better after answering five questions on it than after reading it twice.
- A monomial must be multiplied into every term of a polynomial, and every term must be divided too.
- Two binomials give four products: after collecting like terms.
- One side of a rectangle comes from dividing the area by the other side: .
- For algebraic fractions, use the LCM of the denominators and keep numerator brackets intact — a minus before a bracket reaches every term, so .
- Remove brackets in the order vinculum, ( ), { }, [ ], rewriting the expression at each step.
You will remember all of this far better after answering five questions on it than after reading it twice.