Grade by Grade
C. Polynomial and Rational Functions · Polynomial Functions and Equations

What Is a Polynomial Function?

C1.1About 65 minutes
Learning goals
  • I can decide whether an expression is a polynomial and explain why or why not.
  • I can write a polynomial function in standard form and name its degree, leading coefficient and constant term.
  • I can find the degree and leading coefficient of a product such as (2x − 1)³(x + 4) without expanding it fully.
  • I can build and evaluate a polynomial function that models a real situation and state a sensible domain.

Words to know

polynomial expression
A sum of terms of the form ax^n, where each coefficient a is a real number and each exponent n is a whole number (0, 1, 2, 3, ...).
polynomial function
A function f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ whose rule is a polynomial expression in x.
degree
The highest exponent of x in a polynomial (after like terms are collected).
leading coefficient
The coefficient aₙ of the term with the highest power of x. It is never 0.
constant term
The term a₀ with no x. It is the y-intercept of the graph.
standard form
A polynomial written with its terms in order of decreasing powers of x.
cubic, quartic, quintic
Polynomial functions of degree 3, 4 and 5.

A box from a sheet of cardboard

Take a sheet of cardboard 30 cm long and 20 cm wide. Cut a square with side x out of each corner and fold up the sides. You get an open box. How does its volume depend on x?

The base is (30 − 2x) by (20 − 2x) and the height is x, so

V(x) = x(30 − 2x)(20 − 2x) = 4x³ − 100x² + 600x.

This is a cubic function. It is not linear and not quadratic, but it is built from the same pieces: whole-number powers of x multiplied by numbers and added together. Functions like this are called polynomial functions, and they are the subject of this whole unit.

Remember from Grade 11: in Functions and Equivalent Expressions (MCR3U) you added, subtracted and multiplied polynomials, and in Quadratic Functions you studied degree-2 polynomials in detail. Now we go to any degree.

xxxx30 cm20 cmbase: (30 − 2x) by (20 − 2x)cut squares (x by x), fold up the sides30 − 2xx20 − 2xV(x) = x(30 − 2x)(20 − 2x)
Cutting an x-by-x square from each corner of a 30 cm by 20 cm sheet leaves a base of (30 − 2x) by (20 − 2x). The height of the folded box is x.

What counts as a polynomial?

A polynomial expression in x is a sum of terms. Each term is a real number (the coefficient) times a whole-number power of x: x⁰ = 1, x¹, x², x³, and so on.

These are polynomials:

  • 5x³ − 2x + 1
  • −x⁴ + ½x² − √3 (coefficients may be fractions or irrational numbers)
  • 7 (a constant: 7 = 7x⁰)
  • (x − 2)(x + 5)² (it expands to a sum of whole-number powers)

These are not polynomials:

  • 3/x + 1, because 3/x = 3x⁻¹ has a negative exponent
  • √x + 4, because √x = x^(1/2) has a fractional exponent
  • 2ˣ, because the variable is in the exponent
  • sin x and |x|, which cannot be written as a finite sum of powers of x

A polynomial function is a function whose rule is a polynomial: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, with aₙ ≠ 0. Its domain is all real numbers unless a real situation limits it.

f(x) = −2x⁴ + 5x³ − x + 7leading coefficient: −2highest exponent 4 → degree 4 (quartic)term 5x³term −x (coefficient −1)constant term 7Missing powers (x² here) simply have coefficient 0.
The parts of a polynomial written in standard form (decreasing powers). The degree is the highest exponent; the leading coefficient belongs to that term.
DegreeNameExampleShape you already know?
0constantf(x) = −3horizontal line
1linearf(x) = 2x − 5line (Grade 9 and 10)
2quadraticf(x) = x² − 4x + 1parabola (Grade 10 and 11)
3cubicf(x) = 4x³ − 100x² + 600xnew this unit
4quarticf(x) = −2x⁴ + 5x³ − x + 7new this unit
5quinticf(x) = x⁵ − xnew this unit
Names for polynomial functions by degree.

Degree of a product without expanding

When polynomials are multiplied, their degrees add and their leading coefficients multiply. The constant term of the product is the product of the constant terms. So for f(x) = (2x − 1)³(x + 4): degree 3 + 1 = 4, leading coefficient 2³ × 1 = 8, constant term (−1)³ × 4 = −4. You never had to expand.

Worked example

Example 1: Polynomial or not?

Which of these are polynomial functions? For each one that is, give the degree and leading coefficient.

(a) f(x) = 3 − 4x² + x⁵ (b) g(x) = 2x^(1/2) + x (c) h(x) = (x² + 1)/5 (d) k(x) = 4/x² − 1 (e) m(x) = πx³ − 2ˣ

  1. (a) Every exponent (0, 2, 5) is a whole number. In standard form f(x) = x⁵ − 4x² + 3. Polynomial, degree 5, leading coefficient 1.
  2. (b) x^(1/2) has a fractional exponent. Not a polynomial.
  3. (c) Dividing by the number 5 is fine: h(x) = (1/5)x² + 1/5. Polynomial, degree 2, leading coefficient 1/5.
  4. (d) 4/x² = 4x⁻² has a negative exponent. Dividing by an expression in x is not allowed. Not a polynomial.
  5. (e) πx³ is a fine term (π is just a real number), but 2ˣ has the variable in the exponent. Not a polynomial.

(a) and (c) are polynomial functions; (b), (d) and (e) are not.

Worked example

Example 2: Reading a product

For f(x) = −3x(x − 2)²(2x + 5), find the degree, the leading coefficient and the constant term without expanding.

  1. Degrees of the factors: −3x has degree 1, (x − 2)² has degree 2, (2x + 5) has degree 1. Degree = 1 + 2 + 1 = 4.
  2. Leading coefficients multiply: (−3)(1²)(2) = −6.
  3. Constant term: the factor −3x has constant 0, so the product's constant term is 0. (The graph passes through the origin.)

Degree 4 (quartic), leading coefficient −6, constant term 0. In expanded form f(x) = −6x⁴ + ... + 0.

Worked example

Example 3: Evaluating a polynomial

For f(x) = 2x⁴ − 3x³ + x − 7, find f(−2) and f(1/2).

  1. f(−2) = 2(16) − 3(−8) + (−2) − 7 = 32 + 24 − 2 − 7 = 47. (Be careful: (−2)⁴ = 16 but (−2)³ = −8.)
  2. f(1/2) = 2(1/16) − 3(1/8) + 1/2 − 7 = 1/8 − 3/8 + 4/8 − 56/8 = −54/8 = −27/4.

f(−2) = 47 and f(1/2) = −27/4 = −6.75.

Worked example

Example 4: The box model

Use V(x) = x(30 − 2x)(20 − 2x) from the hook.

(a) Expand V(x) into standard form. (b) Find V(2) and V(5) and explain what they mean. (c) State a sensible domain.

  1. (a) (30 − 2x)(20 − 2x) = 600 − 60x − 40x + 4x² = 4x² − 100x + 600. Multiply by x: V(x) = 4x³ − 100x² + 600x.
  2. (b) V(2) = 2(26)(16) = 832: cutting 2 cm squares gives a box of 832 cm³. V(5) = 5(20)(10) = 1000: cutting 5 cm squares gives 1000 cm³, which is exactly one litre.
  3. (c) Every length must be positive: x > 0 and 20 − 2x > 0, so x < 10. The domain is 0 < x < 10. The formula works for any real x, but only these values make a real box.

V(x) = 4x³ − 100x² + 600x, V(2) = 832 cm³, V(5) = 1000 cm³, domain 0 < x < 10.

x (cm)V (cm³)123456789101120040060080010001200(5, 1000)max ≈ 1056 at x ≈ 3.9
The box volume V(x) = 4x³ − 100x² + 600x on its domain 0 < x < 10. It rises to a largest value of about 1056 cm³ near x ≈ 3.9 cm and falls back to 0 at x = 10. The open dot there (a ring with a lightly shaded centre) shows that x = 10 itself is not in the domain.

Why polynomials matter

Polynomials are the simplest functions that can bend more than once. Engineers use them to describe the shape of roads and roller-coaster tracks, computer graphics uses cubic curves to draw smooth letters on your screen, and in Calculus and Vectors (MCV4U) you will see that polynomials are the easiest functions to differentiate. Many harder functions, such as sin x and eˣ, can even be approximated very closely by polynomials, a trick that calculators and computers often use.

Just for fun

The word polynomial is a mix of two languages: Greek poly ("many") joined to the ending of binomial, which comes from Latin. A word built from terms of different origins, describing an expression built from many terms. Fitting.

Hands-on activity

Build the box

You need: one sheet of letter paper (21.6 cm by 27.9 cm), a ruler, a pencil, tape and scissors (be careful with them; a grown-up should supervise younger siblings who join in).

  1. Write the volume function for this sheet: V(x) = x(27.9 − 2x)(21.6 − 2x).
  2. Choose x = 2 cm. Draw 2 cm squares in the four corners, cut them out, fold up the sides and tape the corners.
  3. Measure the length, width and height of your box and multiply them. Compare with V(2) from your formula.
  4. Make a second box from a new sheet with x = 4 cm and compare V(4) with your measurement.
  5. Which box holds more? Predict whether x = 6 cm would be bigger or smaller, then check with the formula.

What to look for: V(2) = 2(23.9)(17.6) ≈ 841 cm³, V(4) = 4(19.9)(13.6) ≈ 1083 cm³ and V(6) = 6(15.9)(9.6) ≈ 916 cm³. Measured volumes should be within a few percent. The volume rises and then falls, which is the bend that a straight line cannot make.

Your turn

1

Is f(x) = 5x³ − x^(−2) + 1 a polynomial function? Explain.

2

Write g(x) = 4 − x + 3x⁴ − 2x² in standard form and give its degree and leading coefficient.

3

Give the degree, leading coefficient and constant term of f(x) = (x + 3)²(1 − 2x).

4

Evaluate p(x) = x³ − 4x² + 2x − 1 at x = −1 and x = 3.

5

Is h(x) = (x² − 9)/3 a polynomial? Is k(x) = 3/(x² − 9)?

6

Name the type of function: (a) y = 7 − x (b) y = x⁴ − 1 (c) y = 2x³ + x⁵ (d) y = 12

7

A 40 cm by 40 cm sheet has x-by-x squares cut from the corners to make an open box. Write V(x) and state its domain.

More practice: warm-up

1

What is the degree of 6x² − x⁷ + 2?

2

What is the leading coefficient of −x⁵ + 4x³?

3

What is the constant term of 2x³ − 5x + 8?

4

Is √2·x³ + x a polynomial?

5

Is 4ˣ + x a polynomial?

More practice: on your own

1

Find the degree and leading coefficient of f(x) = (3x² − 1)(2 − x)³.

2

For f(x) = −x⁴ + 2x³ − 5, find f(2) and f(−2).

3

Expand and simplify f(x) = (x − 1)(x + 2)(x − 3) into standard form.

4

Explain why g(x) = |x| is not a polynomial, even though it equals x for x ≥ 0.

5

A cylindrical can has height 3 cm more than its radius r. Write its volume V(r) as a polynomial in standard form and give the degree.

Challenge

1

Find the coefficient of x³ in (x + 2)⁴ without expanding all of it. (Hint: choose which factors give an x.)

2

f(x) = ax³ + bx² + cx + d has f(0) = 4 and f(1) = 10. What can you say about d and about a + b + c?

3

A box is made from a 30 cm by 20 cm sheet as in the lesson, but this time the box has a lid: squares are cut and the sheet is folded so that the volume is V(x) = x(20 − 2x)(30 − 2x)/2. What is the degree and leading coefficient, and what is V(4)?

Check your understanding

Quiz

1.

Which of these is a polynomial function?

2.

What is the degree of f(x) = (x − 4)³(2x + 1)²?

3.

What is the leading coefficient of f(x) = −2(x − 1)(3x + 2)²?

4.

For f(x) = x⁴ − 2x³ + x − 5, what is f(−1)?

5.

Why is g(x) = 5x⁻¹ + 2 not a polynomial?

6.

A box is made from a 24 cm by 24 cm sheet by cutting x-by-x squares from the corners. Which is the volume function and domain?

For parents and teachers

Big idea (C1.1): a polynomial is a sum of terms, each a number times a whole-number power of x. The degree, leading coefficient and constant term tell a lot about the function, and can often be read from a factored form without expanding.

How to help: Write a few expressions on scraps of paper (x³ − 2x, 1/x, 2ˣ, √x, (x + 1)²/3) and ask your teen to sort them into polynomials and non-polynomials, giving the reason for each "no". Then ask for the degree of a product such as (x − 1)²(x + 3)³: the answer (5) should come quickly, without expanding.

Common difficulties:

  • Thinking that dividing by a number (x²/3) is not allowed; only dividing by an expression in x breaks the rule.
  • Sign errors when evaluating negative inputs: (−2)⁴ = 16 but −2⁴ = −16, and (−2)³ = −8.
  • Forgetting that the leading coefficient of a factor like (2 − x) is −1, not 2.
  • Giving a model a domain of "all real numbers" when the real situation (a box, a length) needs positive values.