Only Like Terms Can Ever Be Added Together
Learn the vocabulary of algebraic expressions, how to spot like terms, how to add and subtract expressions by collecting them, and how to substitute negative values safely.
Why is $3x + 2y$ as far as you can go?
Because and are different things, and adding them is like adding three pens to two notebooks — you cannot report the total as five of anything.
But does simplify, to , because both terms count the same unknown. Three lots of plus two lots of is five lots of , whatever happens to be.
That single distinction — like terms against unlike terms — governs every simplification in this chapter. Nothing else in algebraic addition is hard; the whole difficulty is recognising which terms may be combined.
This page covers the first part of the ICSE Class 8 Mathematics chapter on algebraic expressions.
But does simplify, to , because both terms count the same unknown. Three lots of plus two lots of is five lots of , whatever happens to be.
That single distinction — like terms against unlike terms — governs every simplification in this chapter. Nothing else in algebraic addition is hard; the whole difficulty is recognising which terms may be combined.
This page covers the first part of the ICSE Class 8 Mathematics chapter on algebraic expressions.
Formula
What are the terms, coefficients and degree of an expression?
Terms are the parts of an expression separated by or signs. Consider
This has four terms: , , and .
A term carries its own sign. The second term is , not , and writing it without the minus sign is the commonest source of error in the whole chapter.
Coefficient is the numerical factor of a term. Here the coefficients are , , , and the last term is a constant — it has no variable at all.
Degree of a term is the sum of the exponents of its variables:
- has degree
- has degree
- has degree
- has degree
Degree of the expression is the highest of these, so this expression has degree .
Naming expressions by their number of terms:
- Monomial — one term, such as
- Binomial — two terms, such as
- Trinomial — three terms, such as
- Polynomial — any number of terms
A hidden coefficient of one. In the first coefficient is , not nothing. Writing is shorthand for , and that matters the moment you add or subtract the term.
A term with a fractional coefficient is still a perfectly ordinary term: in the coefficients are and .
Worked example — full description. Describe .
Three terms, so a trinomial. Coefficients , and the constant . Degrees , and , so the expression has degree .
This has four terms: , , and .
A term carries its own sign. The second term is , not , and writing it without the minus sign is the commonest source of error in the whole chapter.
Coefficient is the numerical factor of a term. Here the coefficients are , , , and the last term is a constant — it has no variable at all.
Degree of a term is the sum of the exponents of its variables:
- has degree
- has degree
- has degree
- has degree
Degree of the expression is the highest of these, so this expression has degree .
Naming expressions by their number of terms:
- Monomial — one term, such as
- Binomial — two terms, such as
- Trinomial — three terms, such as
- Polynomial — any number of terms
A hidden coefficient of one. In the first coefficient is , not nothing. Writing is shorthand for , and that matters the moment you add or subtract the term.
A term with a fractional coefficient is still a perfectly ordinary term: in the coefficients are and .
Worked example — full description. Describe .
Three terms, so a trinomial. Coefficients , and the constant . Degrees , and , so the expression has degree .
How do you identify like terms?
Like terms have exactly the same variables raised to exactly the same powers. The coefficients may differ — indeed they usually do — but the variable part must match completely.
Like:
- and — both plain
- and — multiplication order does not matter, so both are
- and — same variables, same powers
- and — both constants
Unlike:
- and — different variables
- and — same variable, different powers
- and — the power of differs
- and — one has a variable, one does not
The test that settles every case. Cover the coefficients with your finger. If what remains is identical, the terms are like. and both leave ; and leave and , which are not the same.
**Why and can never be combined.** Try : then and , which are different numbers. Try : and . Since they take different values for almost every , there is no single term that could stand for their sum. is already in its simplest form.
Worked example — grouping a long expression. Group the like terms in
- terms: and
- terms: and
- terms: and
- constants:
Three groups combine and one stands alone, which is exactly how a simplification should be set out before any arithmetic happens.
Like:
- and — both plain
- and — multiplication order does not matter, so both are
- and — same variables, same powers
- and — both constants
Unlike:
- and — different variables
- and — same variable, different powers
- and — the power of differs
- and — one has a variable, one does not
The test that settles every case. Cover the coefficients with your finger. If what remains is identical, the terms are like. and both leave ; and leave and , which are not the same.
**Why and can never be combined.** Try : then and , which are different numbers. Try : and . Since they take different values for almost every , there is no single term that could stand for their sum. is already in its simplest form.
Worked example — grouping a long expression. Group the like terms in
- terms: and
- terms: and
- terms: and
- constants:
Three groups combine and one stands alone, which is exactly how a simplification should be set out before any arithmetic happens.
How do you add and subtract algebraic expressions?
Collect the like terms and add their coefficients. The variable part never changes.
Worked example 1 — simplifying. Simplify the expression grouped above:
Four terms remain, and the result has degree .
Worked example 2 — adding two expressions. Add and .
Worked example 3 — subtracting. Subtract from .
The crucial step is changing the sign of every term being subtracted:
Notice became . Forgetting to change the sign of the last term is the single most frequent mistake in subtraction, because the first term's sign is obvious and the later ones are not.
Worked example 4 — a check by substitution. Put into the original problem and into the answer.
First expression: . Second: . Their difference is .
The answer at : . The two agree, so the subtraction was carried out correctly.
This check costs two lines and catches almost every sign error. It is not a proof — an answer could agree at one value by coincidence — but a disagreement is certain evidence of a mistake, and agreement at and together is very strong.
Worked example 5 — finding a missing expression. What must be added to to give ?
Subtract the first from the second:
Here became , since subtracting a negative term adds it.
Worked example 1 — simplifying. Simplify the expression grouped above:
Four terms remain, and the result has degree .
Worked example 2 — adding two expressions. Add and .
Worked example 3 — subtracting. Subtract from .
The crucial step is changing the sign of every term being subtracted:
Notice became . Forgetting to change the sign of the last term is the single most frequent mistake in subtraction, because the first term's sign is obvious and the later ones are not.
Worked example 4 — a check by substitution. Put into the original problem and into the answer.
First expression: . Second: . Their difference is .
The answer at : . The two agree, so the subtraction was carried out correctly.
This check costs two lines and catches almost every sign error. It is not a proof — an answer could agree at one value by coincidence — but a disagreement is certain evidence of a mistake, and agreement at and together is very strong.
Worked example 5 — finding a missing expression. What must be added to to give ?
Subtract the first from the second:
Here became , since subtracting a negative term adds it.
How do you evaluate an expression at negative values?
Substitute in brackets. Almost every error in evaluation comes from omitting them.
Worked example 1. Find the value of when and .
Substituting with brackets:
Term by term:
-
-
-
Where the brackets earn their keep. , but means . The bracket says the minus sign is part of the number being squared; without it, the minus applies only after squaring. These give answers of opposite sign, and the difference is invisible unless the brackets are written.
Worked example 2 — a longer evaluation. Find the value of at , .
-
-
-
-
The last term needed two sign changes: because an odd power keeps the negative sign, and the leading minus then flipped it to .
The rule worth memorising. A negative number raised to an even power is positive; raised to an odd power it stays negative. So but .
Worked example 3 — evaluating a known identity. Find when and .
The middle term became because with negative gives .
Now notice that is , and
The same answer by a much shorter route. Recognising an expression as an identity often turns a page of substitution into one line — which is the whole point of the identities that follow in the next part of the chapter.
Worked example 1. Find the value of when and .
Substituting with brackets:
Term by term:
-
-
-
Where the brackets earn their keep. , but means . The bracket says the minus sign is part of the number being squared; without it, the minus applies only after squaring. These give answers of opposite sign, and the difference is invisible unless the brackets are written.
Worked example 2 — a longer evaluation. Find the value of at , .
-
-
-
-
The last term needed two sign changes: because an odd power keeps the negative sign, and the leading minus then flipped it to .
The rule worth memorising. A negative number raised to an even power is positive; raised to an odd power it stays negative. So but .
Worked example 3 — evaluating a known identity. Find when and .
The middle term became because with negative gives .
Now notice that is , and
The same answer by a much shorter route. Recognising an expression as an identity often turns a page of substitution into one line — which is the whole point of the identities that follow in the next part of the chapter.
Exam tip
Exam tip: bracket every substitution and every subtraction
Combine only like terms — identical variables to identical powers. Cover the coefficients; if what remains matches, they are like. and are like; and are not.
When subtracting an expression, change the sign of every term inside the bracket, not just the first. Missing the last term's sign is the classic slip.
When substituting, always write the value in brackets. while , and no working shows which you meant unless the brackets are there.
Even powers of a negative number are positive; odd powers stay negative.
A term includes its sign: in the second term is .
Remember the **invisible coefficient of **: in the first coefficient is .
Degree of a term is the sum of its exponents ( has degree ); degree of the expression is the highest term degree. A constant has degree .
And **check by substituting ** into both the question and your answer. Two lines of arithmetic, and a mismatch proves an error beyond doubt.
When subtracting an expression, change the sign of every term inside the bracket, not just the first. Missing the last term's sign is the classic slip.
When substituting, always write the value in brackets. while , and no working shows which you meant unless the brackets are there.
Even powers of a negative number are positive; odd powers stay negative.
A term includes its sign: in the second term is .
Remember the **invisible coefficient of **: in the first coefficient is .
Degree of a term is the sum of its exponents ( has degree ); degree of the expression is the highest term degree. A constant has degree .
And **check by substituting ** into both the question and your answer. Two lines of arithmetic, and a mismatch proves an error beyond doubt.
Did you know
Why do we bother with letters at all?
Because a letter lets you write down what you do not yet know and then reason about it anyway.
Before algebraic notation settled into its modern form, problems of this kind were written out in sentences — the thing plus three times the thing gives twenty — and every problem had to be solved afresh in words. The reasoning existed, but it could not be manipulated on the page.
Once the unknown became a symbol, something remarkable followed: the same rules of arithmetic that govern numbers turned out to govern the symbols too. You may add and for exactly the reason you may add three metres and two metres, and the licence to rearrange into is the distributive law you already knew.
That is why nothing in this chapter is a new rule. Like terms combine because addition is what it always was. Signs distribute over a bracket because subtraction is what it always was.
Algebra is not a second system of arithmetic. It is arithmetic performed on things whose values you have not yet been told.
Before algebraic notation settled into its modern form, problems of this kind were written out in sentences — the thing plus three times the thing gives twenty — and every problem had to be solved afresh in words. The reasoning existed, but it could not be manipulated on the page.
Once the unknown became a symbol, something remarkable followed: the same rules of arithmetic that govern numbers turned out to govern the symbols too. You may add and for exactly the reason you may add three metres and two metres, and the licence to rearrange into is the distributive law you already knew.
That is why nothing in this chapter is a new rule. Like terms combine because addition is what it always was. Signs distribute over a bracket because subtraction is what it always was.
Algebra is not a second system of arithmetic. It is arithmetic performed on things whose values you have not yet been told.
Key takeaways
Algebraic expressions and like terms: quick revision
- Terms are separated by or , and each term carries its own sign. In there are four terms, the second being .
- Coefficient is the numerical factor; a term with no variable is a constant. In the first coefficient is an invisible .
- Degree of a term is the sum of its exponents: has degree , has degree , a constant has degree . Degree of the expression is the highest — here .
- Monomial (one term), binomial (two), trinomial (three), polynomial (any number).
- Like terms have identical variables to identical powers. and are like; and are not — at they are and .
- Simplify by adding coefficients: .
- Add: .
- Subtract by changing the sign of every term: .
- **Check at **: and . Agreement is strong evidence; disagreement is proof of error.
- Missing expression: what added to gives ? Subtract to get .
- Substitute in brackets. At , : .
- but . Even powers of a negative are positive, odd powers stay negative, so .
- Longer case: at , gives .
- Spotting an identity shortens the work: at , is , and equals .
Practise one subtraction and one negative substitution every day for a week — those two operations account for most of the marks lost in this chapter, and both are habits rather than difficulties.
- Coefficient is the numerical factor; a term with no variable is a constant. In the first coefficient is an invisible .
- Degree of a term is the sum of its exponents: has degree , has degree , a constant has degree . Degree of the expression is the highest — here .
- Monomial (one term), binomial (two), trinomial (three), polynomial (any number).
- Like terms have identical variables to identical powers. and are like; and are not — at they are and .
- Simplify by adding coefficients: .
- Add: .
- Subtract by changing the sign of every term: .
- **Check at **: and . Agreement is strong evidence; disagreement is proof of error.
- Missing expression: what added to gives ? Subtract to get .
- Substitute in brackets. At , : .
- but . Even powers of a negative are positive, odd powers stay negative, so .
- Longer case: at , gives .
- Spotting an identity shortens the work: at , is , and equals .
Practise one subtraction and one negative substitution every day for a week — those two operations account for most of the marks lost in this chapter, and both are habits rather than difficulties.