 # 4.08 Polynomials

Lesson

## Introduction

An expression of the form Ax^{n}, where A is any number and n is any non-negative integer, is called a monomial. When we take the sum of multiple monomials, we get a polynomial.

## Parts of polynomials

In the monomial Ax^{n}:

• A is the coefficient

• x is the variable

• n is the index

A polynomial is a sum of any number of monomials (and consequently, each term of a polynomial is a monomial). The highest index is called the degree of the polynomial. For example, x^{3}+4x+3 is a polynomial of degree three. The coefficient of the term with the highest index is called the leading coefficient. The coefficient of the term with index 0 is called the constant.

We often name polynomials using function notation. For example P(x) is a polynomial where x is the variable. If we write a constant instead of x, that means that we substitute that constant for the variable. For example, if P(x)=x^{3}+4x+3 then P(3)=3^{3}+4\times 3+3=42.

Polynomials of particular degrees are given specific names. Some of these we have seen before.

### Examples

#### Example 1

For the polynomial P(x)=\dfrac{x^7}{5}+\dfrac{x^6}{6}+5.

a

What's the degree of the polynomial?

Worked Solution
Create a strategy

The degree of a polynomial is the highest power of x.

Apply the idea

The highest power of x is 7, so the degree of the polynomial is 7.

b

What's the leading coefficient of the polynomial?

Worked Solution
Create a strategy

The term containing the highest power of x is called the leading term, and its coefficient is the leading coefficient.

Apply the idea

The leading term of the polynomial is \dfrac{x^7}{5} which can be written as \dfrac{1}{5}x^7. So the leading coefficient of the polynomial is \dfrac{1}{5}.

c

What's the constant term of the polynomial?

Worked Solution
Create a strategy

The constant term is the term that is independent of x.

Apply the idea

The term that is independent of x is 5, so the constant term of the polynomial is 5.

#### Example 2

Consider P(x)=4x^5+3x^{6}-8.

a

Find P(0).

Worked Solution
Create a strategy

Substitute x=0 into the polynomial.

Apply the idea
b

Find P(-4).

Worked Solution
Create a strategy

Substitute x=-4 into the equation.

Apply the idea
Idea summary

A monomial is an expression of the form:

\displaystyle Ax^{n}
\bm{A}
is the coefficient
\bm{x}
is the variable
\bm{n}
is the index

A polynomial is a sum of any number of monomials. The highest index is called the degree of the polynomial. The coefficient of the term with the highest index is called the leading coefficient. The coefficient of the term with index 0 is called the constant.

## Operations on polynomials

We apply operations to polynomials in the same way as we apply operations to numbers. For addition and subtraction we add or subtract all of the terms in both polynomials and we simplify by collecting like terms. For multiplication we multiply each term in one polynomial by each term in the other polynomial similar to how we expand binomial products. Division is a more complicated case that we will look at in the  next lesson  .

A polynomial is a sum of any number of monomials. In a polynomial:

• The highest index is the degree

• The coefficient of the term with the highest index is the leading coefficient

• The coefficient of the term with index 0 is the constant

We apply operations to polynomials in the same way that we apply operations to numbers.

### Examples

#### Example 3

If P(x)=-5x^{2}-6x-6 and Q(x)=-7x+7, form a simplified expression for P(x)-Q(x).

Worked Solution
Create a strategy

Substitute the expressions for P(x) and Q(x), and then subtract the like terms.

Apply the idea

#### Example 4

Simplify \left(3x^{3}-9x^{2}-8x-7\right)+\left(-7x^{3}-9x\right).

Worked Solution
Create a strategy

Apply the idea
Idea summary

A polynomial is a sum of any number of monomials. In a polynomial:

• The highest index is the degree

• The coefficient of the term with the highest index is the leading coefficient

• The coefficient of the term with index 0 is the constant

### Outcomes

#### VCMNA357 (10a)

Investigate the concept of a polynomial and apply the factor and remainder theorems to solve problems.