One block held together by multiplication is a monomial. Snap several blocks together with + and − and you get a polynomial. Both belong to the same family of algebraic expressions.
In Lesson 7.1 you learned that a letter can stand for a number, so an expression like 3x2 − 2x + 1 is really a recipe that produces a number once you know what x is. This lesson is about sorting and naming those recipes. Just as a biologist gives every animal a genus and a species, mathematicians give every algebraic expression a precise name — and the names tell you, at a glance, exactly how the expression is built.
By the end you will be able to do five things: recognize a monomial (a single block made only by multiplying), read off its coefficient and its degree, join monomials into a polynomial and count its terms, find a polynomial's degree and its constant term, and finally pick out like terms — the matching blocks you are allowed to combine. We keep one steady habit of color throughout: letters are blue, numbers in front are amber, and exponents are purple.
7.2.1 Monomials: held together only by multiplication
Start with the simplest kind of expression: one in which nothing is ever added or subtracted — everything is just multiplied together. A number, a letter, or a clump of numbers and letters multiplied is called a monomial (the prefix mono means "one"). Think of it as a single, unbreakable building block.
Here are four genuine monomials: 3x2, −2ab, 7, and just x. Look closely at each. In 3x2 the 3 multiplies x2; in −2ab the −2 multiplies a and b; the lone number 7 is a monomial all by itself; and a single letter x is the leanest block of all. Crucially, not one of them contains a plus or a minus sign inside. (The minus in −2ab is not an operation between two pieces — it just makes the number out front negative.)
A monomial is one block. The moment a + or a − separates two pieces, you no longer have a single monomial — you have several joined together.
Key idea
A monomial is a number, a letter, or numbers and letters multiplied together — and nothing else. No plus, no minus inside. It is one indivisible building block, the atom of algebra.
Watch out — division by a letter is not allowed
A monomial may multiply letters, but it must not divide by a letter. So 5x is a monomial, but 5 ÷ x (that is, 5x) is not. A plain number divisor is fine, though: x/2 is the same as ½x, which is just a number times a letter — a perfectly good monomial.
🎮 Try itInspect a monomial
Build the monomial cxayb. Set the coefficient and the two exponents, and watch the block rebuild itself.
Coefficient−3
Power of x2
Power of y1
7.2.2 Coefficient and degree of a monomial
Every monomial carries two numbers that describe it. The first is easy to see and the second takes a moment of counting.
The coefficient is the number written out front — it tells you the size of the block, how many copies of the letter part you have. In −3x2y, the coefficient is −3. The sign always travels with the coefficient, so the coefficient here is negative three, not three.
The degree measures how many "layers" of letters the block has. To find it, you add up the exponents on all the letters. In −3x2y, the x carries an exponent of 2 and the y carries an exponent of 1 (a letter with no written exponent counts as 1). So the degree is 2 + 1 = 3.
In −3x2y: the coefficient is the number out front, and the degree is the sum of the exponents, 2 + 1 = 3.
Worked example — coefficient and degree of 5a3b2
Find the coefficient and the degree of 5a3b2.
The number out front is 5, so the coefficient is 5. it includes its sign — here, positive
Read each exponent: a has 3, b has 2. every letter carries an exponent
Add them: 3 + 2 = 5, so the degree is 5. degree = sum of all the exponents
Watch out — the three sneaky cases
① A letter alone, like x, has an invisible coefficient of 1 (it means 1x) and degree 1.
② −x has coefficient −1, not 0 and not −1 times nothing — the lone minus means the coefficient is −1.
③ A nonzero number with no letters, like 7, has degree 0 (there are no exponents to add, so the sum is 0).
🎮 Try itRead the coefficient and degree
Step a monomial through several presets. The coefficient lights up amber and the exponents add up to the degree, layer by layer.
Monomial1 / 6
7.2.3 Polynomials: several blocks joined
One block is rarely enough to describe something interesting. So we snap monomials together using the only two signs that join, not multiply: plus and minus. Link a few monomials with + and − and you have built a polynomial (the prefix poly means "many"). For example, the three blocks x2, 2x, and 1 joined as x2 − 2x + 1 form a single polynomial.
Each block in the chain is called a term. The polynomial x2 − 2x + 1 has three terms: x2, then −2x, then 1. Notice the second term is −2x, with its minus sign attached. This is the single most important habit in all of polynomials:
A polynomial is a chain of monomials. Each link is a term — and the sign in front of a term is part of that term, so the middle term here is −2x, not 2x.
Key idea — the sign belongs to the term
To break a polynomial into its terms, snap it apart at the + and − signs, and keep each sign glued to the term that follows it. 4x3 − 7x + 2 has the three terms 4x3, −7x, and +2.
Worked example — listing the terms
List the terms of 2a2 − a − 6.
Find the joining signs: there is a − before a and a − before 6. break only at + and −
Cut there, keeping each sign with its term. the sign rides along
The terms are 2a2, −a, and −6 — three terms in all. −a means −1a
Watch out — a single monomial is also a polynomial
"Poly" means "many," but a chain of one link still counts. Every monomial like 5x is also a (very short) polynomial. So the polynomials are the big family, and the monomials are the one-term members inside it.
🎮 Try itBuild a polynomial from blocks
Add term blocks, choose each one's sign and degree, then delete any you don't want. The polynomial assembles itself in standard form (highest degree first), with each term labeled.
7.2.4 A polynomial's terms, degree, and constant term
Once you can see the terms, three useful facts pop out. They all come from looking at the terms one at a time.
Number of terms. Simply count the links in the chain. 3x2 − 5x + 4 has three terms; x4 + 1 has two.
Degree of the polynomial. Each term has its own degree (the sum of its exponents, from Section 7.2.2). The polynomial's degree is the biggest of those term-degrees — the tallest block in the stack. In 3x2 − 5x + 4, the term degrees are 2, 1, and 0, so the polynomial has degree 2.
Constant term. The term with no letters at all — just a plain number — is the constant term. In 3x2 − 5x + 4 the constant term is 4. (If there is no bare number, the constant term is 0.)
It is good manners — and it makes the degree obvious — to write a polynomial in standard form: terms in order of descending degree, highest power first, the constant last. So you would tidy 4 − 5x + 3x2 into 3x2 − 5x + 4.
Line the terms up by degree, tallest first. The tallest term sets the polynomial's degree (here 2), and the plain-number term is the constant term (here 4).
Worked example — describe 7x − 2x3 + 5
How many terms does 7x − 2x3 + 5 have? What is its degree, its constant term, and its standard form?
Break at the signs: terms are 7x, −2x3, 5 → 3 terms. count the links
Term degrees are 1, 3, and 0; the biggest is 3 → degree 3. tallest block wins
The bare number is 5 → constant term 5. no letters at all
Reorder by descending degree: −2x3 + 7x + 5. highest power first, constant last
Watch out — degree is the biggest, not the sum
A polynomial's degree is the largest term-degree, never the total. In x2 + x + 1 the degree is 2 (the biggest), not 2 + 1 + 0 = 3. And a degree of 3 needs an actual term of degree 3 — it does not come from adding up smaller terms.
🎮 Try itDegree & terms detector
A polynomial appears. Set your guesses for its degree and its number of terms, then check. Cycle through several and try to get them all right.
Degree0
# terms1
7.2.5 The family and like terms
Step back and look at the whole household. Monomials are the single blocks; polynomials are several blocks joined. Together they make the family of algebraic expressions we will work with for the rest of Stage 7. The monomials are simply the one-term members of the polynomial family.
Now for the idea that powers nearly everything to come. Two terms are called like terms when they have exactly the same letters raised to exactly the same exponents. Only the number out front may differ. Like terms are matching blocks — interchangeable except for how many you have — and matching blocks are the only ones you are ever allowed to add or subtract together (you will do exactly that in Lesson 7.3).
So 3x2 and −5x2 are like terms: both are "x-squared" blocks, differing only in the amber number. But 3x2 and 3x are not like terms — one is an x-squared block and the other is an x block, a different shape entirely, even though the coefficient is the same.
Like terms must match in both the letters and their exponents. A bigger or different coefficient is fine; a different exponent makes them a different kind of block.
Key idea — the like-terms test
Cover up the coefficients with your thumb. If the letter parts that remain are identical (same letters, same exponents), the terms are like. If anything about the letter part differs, they are unlike. The constant terms (plain numbers) are all like one another, since each has the same empty letter part.
Worked example — group the like terms
From 2x2, −x, 5, 3x2, 4x, −1, sort the like terms into groups.
Cover the coefficients and read the letter parts: x2, x, (none), x2, x, (none). the letter part is what matters
Same letter part = same group. match the shapes
Groups: x²-terms{2x2, 3x2}; x-terms{−x, 4x}; constants{5, −1}. three families
🎮 Try itLike-terms matcher
Click a chip from the pool to drop it into the matching group. Like terms snap together by color; an unlike chip is bounced back. Empty all three groups correctly to win.
Pool — tap a chip
x²-termsx-termsconstants
★ The big ideas, in one breath
A monomial is one block — numbers and letters multiplied, no + or − inside. Its coefficient is the number out front, and its degree is the sum of its exponents (a bare nonzero number has degree 0). Join monomials with + and − and you get a polynomial, whose pieces are terms (each sign belongs to its term). A polynomial's degree is the biggest term-degree, and its constant term is the plain number; write it in standard form, descending degree. Finally, like terms share the same letters with the same exponents — the matching blocks you'll soon learn to combine.
Coming up next — 7.3
Now that you can spot like terms, you are ready to add and subtract expressions: combine the matching blocks, clear away brackets, and tidy a long expression into its shortest standard form. The whole move rests on the like-terms idea you just built.
✎ Practice 7.2
Work each one out first, then open the answer to check your thinking.
Which of these are monomials? 4x2, x + 3, −6, 2ab2, y − 1.
Show answer
Monomials: 4x2, −6, and 2ab2. The other two, x + 3 and y − 1, each contain a + or − between pieces, so they are not single blocks.
Give the coefficient and the degree of −7x3.
Show answer
Coefficient −7 (sign included); degree 3 (the only exponent is 3).
Find the coefficient and degree of 4a2b3.
Show answer
Coefficient 4; degree 2 + 3 = 5.
What is the coefficient and degree of −xy? And of the plain number 9?
Show answer
−xy has coefficient −1 and degree 1 + 1 = 2. The number 9 has coefficient 9 and degree 0 (no letters to give exponents).
List the terms of 5x2 − x + 8, and say how many there are.
Show answer
Terms: 5x2, −x, +8 — three terms. (Remember the second term carries its minus sign.)
For 2x3 − 4x2 + 6, give the number of terms, the degree, and the constant term.
Show answer
3 terms; degree 3 (the biggest term-degree); constant term 6.
Rewrite 5 − 3x2 + x in standard form.
Show answer
−3x2 + x + 5. Order by descending degree (2, then 1, then 0), keeping each sign with its term.
True or false: the degree of x2 + x + 1 is 3. Explain.
Show answer
False. The degree is the biggest term-degree, which is 2 — not the sum 2 + 1 + 0. Don't add the degrees together.
Which pairs are like terms? (a) 4x2 & −9x2 (b) 2x & 2x2 (c) 7 & −3 (d) 5ab & 5a.
Show answer
Like: (a) both are x²-terms, and (c) both are constants. Unlike: (b) exponents 1 vs 2 differ; (d) letters ab vs a differ.
Sort into like-term groups: 3y, −2y2, 7, 5y, y2, −4.