4.07 Polynomials
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.
| Degree | Name |
|---|---|
| Zero | Constant |
| One | Linear |
| Two | Quadratic |
| Three | Cubic |
Examples
Example 1
For the polynomial P(x)=\dfrac{x^7}{5}+\dfrac{x^6}{6}+5.
What's the degree of the polynomial?
What's the leading coefficient of the polynomial?
What's the constant term of the polynomial?
Example 2
Consider P(x)=4x^5+3x^{6}-8.
Find P(0).
Find P(-4).
A monomial is an expression of the form:
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).
Example 4
Simplify \left(3x^{3}-9x^{2}-8x-7\right)+\left(-7x^{3}-9x\right).
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