Math 9 · Polynomials
Like terms, and nothing else adds
A polynomial is an expression built from terms like 3x², −5x and 7. The whole unit rests on one rule — you can only add or subtract like terms — and on the distributive property for multiplying and dividing by a single term. Algebra tiles make both visible.
- 1. The words, first
- 2. Terms, degree and like terms
- 3. Adding and subtracting
- 4. Multiplying and dividing by a monomial
- 5. What costs marks
The words, first
The idea: A term is one piece of an expression, and like terms are pieces of the same kind.
| Word | What it means |
|---|---|
| Term | A number, a variable, or a product of them. Terms are separated by + and −, so 3x² − 5x + 7 has three. |
| Coefficient | The number multiplying the variable, sign included: 3 in 3x², −5 in −5x. |
| Constant term | A term with no variable, like the 7. |
| Monomial, binomial, trinomial | One term (4x), two terms (x + 3) or three terms (3x² − 5x + 7). All are polynomials. |
| Degree | Of a term, the total of the exponents on its variables: 4x² and 3xy both have degree 2. Of a polynomial, the highest degree of its terms. Grade 9 stops at 2. |
| Like terms | Terms with exactly the same variables raised to exactly the same exponents. 3x and −7x are like terms; 3x and 3x² are not. |
| Algebra tiles | Area models: a large square for x², a long strip for x, a small square for 1. One colour is positive, another negative. |
| Distributive property | a(b + c) = ab + ac: multiply the outside term by every term inside the bracket. |
Terms, degree and like terms
The idea: Like terms have identical variable parts. Combining them changes the coefficient and nothing else.
Reading a polynomial. 3x² − 5x + 7 is a trinomial of degree 2. Its coefficients are 3 and −5 — the sign belongs to the term after it — and its constant term is 7.
Why x and x² cannot combine. As tiles, x is a strip and x² is a big square. Three strips and two squares stay three strips and two squares; there is no single tile they become. So 3x + 2x² is finished, and writing 5x³ adds together things of different kinds.
Worked example. Simplify 4x² + 3x − 2 − x² + 5x + 6.
- Group like terms, each carrying its sign: (4x² − x²) + (3x + 5x) + (−2 + 6).
- Combine the coefficients: 3x² + 8x + 4.
- Check with x = 1: 4 + 3 − 2 − 1 + 5 + 6 = 15, and 3 + 8 + 4 = 15 ✓.
Two variables. 2xy and −5yx are like terms, because xy and yx are the same product: together they make −3xy.
Adding and subtracting
The idea: Adding is collecting like terms. Subtracting is adding the opposite — so every sign in the second bracket changes, not only the first.
Adding. (2x² − 3x + 1) + (x² + 5x − 4) = 3x² + 2x − 3. Drop the brackets and collect like terms.
Subtracting. With tiles, subtracting a polynomial means adding its opposite: flip every tile in the second group to the other colour. In symbols, the minus sign in front of the bracket changes the sign of every term inside.
Worked example. (3x² − 2x + 5) − (x² − 4x − 1)
= 3x² − 2x + 5 − x² + 4x + 1 = 2x² + 2x + 6
Check with x = 1. The original is (3 − 2 + 5) − (1 − 4 − 1) = 6 − (−4) = 10, and the answer gives 2 + 2 + 6 = 10 ✓. Changing only the first sign gives 2x² − 6x + 4, which is 0 at x = 1, so the check catches it at once.
Multiplying and dividing by a monomial
The idea: The monomial multiplies every term inside the bracket, or divides every term. The exponent laws from Unit 2 take care of the variables.
Monomial times monomial. Multiply the coefficients and use the product law: (4x)(−2x) = −8x², because x × x = x². Not 2x — that is x + x.
Monomial times polynomial.
−3x(2x − 5) = (−3x)(2x) + (−3x)(−5) = −6x² + 15x
The second product is a negative times a negative, so it is positive. Losing that sign is the usual error.
Dividing: split it over every term.
(6x² − 9x) ÷ 3x = 6x² ÷ 3x − 9x ÷ 3x = 2x − 3
Check by multiplying back: 3x(2x − 3) = 6x² − 9x ✓. Multiplying and dividing undo each other, so this check is always there.
Dividing by a negative changes every sign: (12x² − 8x + 4) ÷ (−4) = −3x² + 2x − 1.
Math 10C multiplies two binomials, like (x + 2)(x + 5), and then runs that backwards as factoring. This page is the first half of that.
What costs marks
The idea: Checking with x = 1 catches four of these five.
- Combining unlike terms. 3x + 2x² is already simplified.
- x + x = x². It is 2x; x × x is x².
- Changing only the first sign when subtracting a bracket.
- Multiplying or dividing only the first term. 2(x + 5) is 2x + 10, not 2x + 5.
- Dropping a negative when the monomial outside the bracket is negative.