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Wize Grade 11 Mathematics Textbook > Polynomials and Operations

Polynomials

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Polynomials

A polynomial function is a finite sum of terms where the variable is raised to a whole number and all the coefficients are real numbers.

More specifically they take on the form:
f(x)=anxn+an−1xn−1+⋯+ax+a0f(x)=a_nx^n + a_{n-1}x^{n-1} + \cdots + ax + a_0
  • The aa's are all real numbers
  • The nn's are whole numbers
  • There are a finite number of terms
Example 1

The following functions are all polynomials

  • f(x)=5x2−3x+1f(x) = 5x^2 - 3x + 1
  • g(x)=7g(x) = 7
Most of the terms are zero except for the constant term. This still meets all the requirements for a polynomial.

  • k(x)=x(x−2)(x+1)k(x)=x(x-2)(x+1)
To see that this is in the proper form it must be multiplied out.

k(x)=x(x−2)(x+1)=x(x2−1x−2)=x3−x2−2x\begin{aligned} k(x) &=x(x-2)(x+1) \\ &= x(x^2 - 1x - 2) \\ &= x^3 - x^2 - 2x \end{aligned}

Wize Tip
It is often easier to determine if an equation is NOT a polynomial function by checking if it violates any of the criteria. It is not a polynomial function if it
  • contains imaginary numbers
  • has fractional exponents or roots
  • divides by the variable
  • contains infinite terms

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Describing polynomials

There are several key words that are used when describing polynomials
  • The leading term of a polynomial is the term with the highest power
  • The degree of a polynomial is the highest power
  • The leading coefficient is the coefficient associated with the leading term
Polynomials are grouped together by their highest degree, or by the number of terms they contain.

Example 2




Wize Tip
To better read the information of a polynomial it is best to write the terms in descending order of their power.

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Example: Polynomials


A piece of cardboard is being made into a box with an open top. This is done by cutting 3 cm squares out of each of the corners, and then folding it up. The cardboard is twice as long as it is wide.


The function that describes the volume of the finished box is given by
f(x)=6x2−54x+108f(x) = 6x^2 - 54x + 108

1. What kind of function this is?

This function is a polynomial. We could describe it as a quadratic or a trinomial.
  • Quadratic since its highest power is 2
  • Trinomial since it has 3 terms

2. What are some of its key features like leading term, degree, and leading coefficient?
  • Leading coefficent: 66
  • Degree: 2
  • Leading term: 6x26x^2

Practice: Polynomials

Only one of the options is a polynomial. Select the polynomial.

Practice: Polynomials

Three polynomials are given below. Identify some key information for each of the polynomials.

1. y=x3+2x4y=x^3 + 2x^4
What is the leading term?

2. y=−3x2+2x−7y=-3x^2 +2x - 7
What is the leading coefficient?

3. y=5y=5
What is the degree?


Practice: Polynomials

Write the equation of a polynomial function with the following criteria.
  • The polynomial is considered a trinomial, so it has only 3 terms.
  • The degree is 2
  • The leading coefficient is -3
  • The coefficient on the term with an xxis 5
  • f(0)=2f(0) = 2