Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Adding, Subtracting and Multiplying Polynomials

A3.1About 70 minutes
Learning goals
  • I can add and subtract polynomials by collecting like terms.
  • I can multiply polynomials, including a binomial by a trinomial and products of three binomials.
  • I can simplify expressions that combine several operations.
  • I can write polynomial expressions for real situations such as volumes.

Words to know

polynomial
An expression made of terms with whole-number exponents on the variables, such as 4x³ − x + 7.
degree
The highest exponent of the variable in a polynomial.
like terms
Terms with exactly the same variables and exponents.
distributive property
a(b + c) = ab + ac. Used to multiply every term by every term.
expand
Multiply out the brackets.
simplify
Rewrite an expression in an equivalent, shorter form.

Why algebra again?

In Grade 9 you collected like terms, used the distributive property and multiplied binomials with tiles and area models. In Grade 10 you expanded and factored second-degree expressions. In this course the expressions get longer (third degree and beyond) because functions and real problems need them. A box's volume, for example, multiplies three lengths. This lesson pulls the old skills together and stretches them.

Vocabulary check

  • A term is a number, a variable, or a product of them: 5, x, −3x²y.
  • A polynomial is a sum of terms with whole-number exponents on the variables: 4x³ − x + 7.
  • The degree of a polynomial in one variable is its highest exponent: 4x³ − x + 7 has degree 3.
  • Like terms have exactly the same variables with the same exponents: 3x² and −8x² are like; 3x² and 3x are not.

Two expressions are equivalent if they give the same value for every value of the variable. Simplifying turns an expression into a shorter, equivalent one.

Adding and subtracting

To add, remove the brackets and collect like terms.

To subtract, remember the minus sign in front of a bracket multiplies every term inside by −1. Change every sign, then collect like terms.

Worked example

Example 1: Adding and subtracting polynomials

Simplify (a) (3x² − 5x + 2) + (x² + 7x − 9) (b) (4x³ − 2x + 1) − (x³ − 6x² − 2x + 8).

  1. (a) Remove brackets: 3x² − 5x + 2 + x² + 7x − 9.
  2. Collect: (3x² + x²) + (−5x + 7x) + (2 − 9) = 4x² + 2x − 7.
  3. (b) Distribute the −1 to every term of the second polynomial: 4x³ − 2x + 1 − x³ + 6x² + 2x − 8.
  4. Collect: (4x³ − x³) + 6x² + (−2x + 2x) + (1 − 8) = 3x³ + 6x² − 7.
  5. Check (b) with x = 1: original (4 − 2 + 1) − (1 − 6 − 2 + 8) = 3 − 1 = 2; answer 3 + 6 − 7 = 2 ✓.

(a) 4x² + 2x − 7 (b) 3x³ + 6x² − 7.

Multiplying polynomials

The distributive property does all the work: multiply every term of the first polynomial by every term of the second, then collect like terms.

  • A binomial times a trinomial gives 2 × 3 = 6 products before collecting.
  • When you multiply powers of x, add the exponents: x · x² = x³.
  • An area model (a grid) keeps track of all the products.
Worked example

Example 2: Binomial times trinomial

Expand and simplify (x − 2)(x² + 3x − 5).

  1. Multiply x by each term: x³ + 3x² − 5x.
  2. Multiply −2 by each term: −2x² − 6x + 10.
  3. Add the six products: x³ + 3x² − 2x² − 5x − 6x + 10.
  4. Collect like terms: x³ + x² − 11x + 10.
  5. Check with x = 2: the original is 0 × (anything) = 0; the answer is 8 + 4 − 22 + 10 = 0 ✓.

x³ + x² − 11x + 10

x²+3x−5xx³3x²−5x−2−2x²−6x+10x³ + (3x² − 2x²) + (−5x − 6x) + 10= x³ + x² − 11x + 10Same fill =like terms6 boxes =2 × 3 products
The area model for (x − 2)(x² + 3x − 5). Each box is one product. Boxes of the same colour hold like terms.
Worked example

Example 3: Squares and cubes of binomials

Write and simplify an expression for the volume of a cube with edge length 2x + 1.

  1. Volume of a cube = edge³, so V = (2x + 1)³ = (2x + 1)(2x + 1)(2x + 1).
  2. First square: (2x + 1)² = 4x² + 4x + 1. (Check: the middle term is 2 × 2x × 1 = 4x.)
  3. Then multiply by (2x + 1): (4x² + 4x + 1)(2x + 1) = 8x³ + 4x² + 8x² + 4x + 2x + 1.
  4. Collect: V = 8x³ + 12x² + 6x + 1.
  5. Check with x = 1: edge 3, so V = 27; the polynomial gives 8 + 12 + 6 + 1 = 27 ✓.

V = 8x³ + 12x² + 6x + 1. Note: (2x + 1)³ is NOT 8x³ + 1.

Don't distribute exponents over addition

(a + b)² is not a² + b², and (a + b)³ is not a³ + b³. Always write the factors out and multiply. A quick check with a number (such as x = 1) catches this mistake every time.

Worked example

Example 4: Several operations

Simplify 2(x + 3)² − 3(x − 1)(x + 4).

  1. Follow the order of operations: powers and products before the outer multiplication and subtraction.
  2. (x + 3)² = x² + 6x + 9, so 2(x + 3)² = 2x² + 12x + 18.
  3. (x − 1)(x + 4) = x² + 3x − 4, so 3(x − 1)(x + 4) = 3x² + 9x − 12.
  4. Subtract the whole second result: 2x² + 12x + 18 − 3x² − 9x + 12.
  5. Collect: −x² + 3x + 30.
  6. Check with x = 0: 2(9) − 3(−1)(4) = 18 + 12 = 30; answer gives 30 ✓.

−x² + 3x + 30

Worked example

Example 5: An open-top box

A sheet of cardboard is 30 cm by 20 cm. Squares of side x cm are cut from each corner and the sides are folded up to make an open box. Write the volume as a polynomial.

  1. After cutting, the base is (30 − 2x) cm long and (20 − 2x) cm wide, and the height is x cm.
  2. V = x(30 − 2x)(20 − 2x).
  3. First multiply the binomials: (30 − 2x)(20 − 2x) = 600 − 60x − 40x + 4x² = 4x² − 100x + 600.
  4. Then multiply by x: V = 4x³ − 100x² + 600x.
  5. Check with x = 5: the box is 20 by 10 by 5, so V = 1000 cm³. The polynomial gives 500 − 2500 + 3000 = 1000 ✓.
  6. In context, 0 < x < 10 (the width 20 − 2x must stay positive).

V = 4x³ − 100x² + 600x cm³, for 0 < x < 10.

xxxx30 cm20 cmbase(30 − 2x) by (20 − 2x)cut out four x-by-x squares, fold up the sides30 − 2x20 − 2xxV = x(30 − 2x)(20 − 2x)
Cutting four x-by-x squares from a 30 cm by 20 cm sheet gives a box with base (30 − 2x) by (20 − 2x) and height x.
Just for fun

Expanding (x + 1)¹⁰⁰ by hand would take 100 brackets and a lot of snacks. In the Sequences and Series unit of this course you will meet Pascal's triangle, which hands you the answer's coefficients without the snacks.

Hands-on activity

Build the box

You need: a sheet of paper or thin cardboard (a cereal box panel works), a ruler, scissors and tape. Take care with scissors; a grown-up can help.

  1. Measure your sheet. Call the length L and the width W in centimetres.
  2. Write V = x(L − 2x)(W − 2x) and expand it into a polynomial.
  3. Cut a 2 cm square from each corner, fold up and tape the sides.
  4. Measure the box and calculate its volume. Then substitute x = 2 into your polynomial. Do they match?
  5. Try x = 3 on a second sheet if you have one. Which box holds more?

What to look for: for a 28 cm by 21.5 cm sheet (letter size, rounded), V = x(28 − 2x)(21.5 − 2x) = 4x³ − 99x² + 602x. At x = 2: 32 − 396 + 1204 = 840 cm³, and the box measures 24 by 17.5 by 2 = 840 cm³. At x = 3: 108 − 891 + 1806 = 1023 cm³, so the deeper box holds more.

Your turn

1

Simplify (5x² − 3x + 4) + (−2x² + 8x − 1).

2

Simplify (6x³ + x − 2) − (2x³ − 4x² + x + 5).

3

Expand −3x²(2x² − x + 6).

4

Expand (x + 4)(x² − 2x + 3).

5

Expand (3x − 2)².

6

Expand (x + 1)(x + 2)(x + 3).

7

What is the degree of 7x − 2x⁴ + x³?

More practice: warm-up

1

Are 4xy and −xy like terms?

2

Simplify x² · x³.

3

Simplify 2x + 3x² − x + x².

4

Expand 5(x − 7).

5

Expand (x + 6)(x − 6).

More practice: on your own

1

Simplify 4(x − 2)² − (x + 3)(x − 5).

2

Expand (2x − 1)(x² + 5x − 4).

3

Expand (x − 3)³.

4

A rectangle has length 3x + 2 and width x − 4. Write and simplify its area and its perimeter.

5

Expand (x² + x − 1)(x² − x + 1).

Challenge

1

A square photo has side x cm. It sits in a frame that adds 3 cm on every side. Write and simplify the area of the frame alone.

2

Find the values of a and b if (x + a)(x² + bx + 4) = x³ + 5x² + 10x + 12.

3

Show that (n + 1)³ − n³ is always odd when n is a whole number.

Check your understanding

Quiz

1.

Simplify (2x² + 3x − 1) − (x² − 4x + 6).

2.

Expand (x + 2)(x² − x + 3).

3.

Which is equal to (x + 3)²?

4.

The volume of a cube with edge x + 2 is:

5.

How many products do you get before collecting like terms when you multiply a binomial by a trinomial?

6.

What is the degree of (x² + 1)(x³ − x)?

For parents and teachers

Big idea: Expectation A3.1 asks students to simplify polynomial expressions by adding, subtracting and multiplying. Grade 9 and 10 covered binomials; here the products are longer (binomial × trinomial, three binomials, cubes).

How to help: After your teen simplifies, have them substitute a small number (x = 1 or x = 2) into both the original and the answer. If the values differ, there is a mistake. This habit also prepares them for the lesson on equivalent expressions.

Common difficulties:

  • Not distributing the negative to every term when subtracting.
  • Writing (a + b)² = a² + b².
  • Missing one of the products in a longer multiplication (count them: 2 × 3 = 6).
  • Adding exponents when adding terms (x² + x² is 2x², not x⁴).