3.01 Operations with polynomials
Introduction
We discovered how to add, subtract, and multiply polynomials in Algebra 1 lesson  9.01 Adding and subtracting polynomials and lesson  9.02 Multiplying polynomials . This lesson will be a review of the vocabulary and the properties of polynomial operations.
Operations with polynomials
In Algebra 1, we learned the following definitions related to polynomials:
The term which has a fixed value and no variables is called the constant term. The term with highest exponent on the variable is called the leading term, and the exponent of this term is the degree of the polynomial.
The standard form of a polynomial is a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_1 x + a_0, where n is a non-negative integer and each a_i is a coefficient.
Polynomials may be in more than one variable. In this case, the degree of a term will be the sum of the exponents for all variables.
| Polynomial | Degree | |
|---|---|---|
| General example | a_i x^m y^n + a_{i - 1} x^{m - 1}y^{n-1} + \ldots + a_1 xy + a_0 | m+n |
| Numerical example | 3xy^3+2xy+11y+3 | 4 |
Polynomials can also have names specific to the number of terms they have. A monomial is a polynomial with one term. A binomial is a polynomial with two terms. A trinomial is a polynomial with three terms.
Exploration
In the definition of polynomials, the constant coefficients are multiplied to the variables, the variables are raised to non-negative integer powers, and the terms are added and subtracted together. This allows the function to be one, smooth curve with no breaks, holes, sharp turns, or stopping points.
Use technology to graph the following functions:
- y=x^2+2x-1
- y=-3x^{5}-4x^{4}+2x^{3}+x^{2}-0.2x
- y=4x^3+\frac{1}{x}
- y=2\sqrt{x}
- y=|x|
- Determine if the function is a polynomial and explain your answer.
One way to determine if an expression or equation is a polynomial is to examine its graph. The graph of a polynomial is a function with one smooth curve over a continuous domain. A polynomial will not have a negative exponent on a variable, a rational exponent on a variable, or have a variable in absolute value bars.
In Algebra 1, we explored the fact that polynomials are closed under addition, subtraction, and multiplication. This means that the sum, product, or difference of polynomials will also be a polynomial.
Addition: if we assume m<n,
| General example | \left(a_{n}x^{n}+\ldots +a_{0}\right)+\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\a_{n}x^n+\ldots +\left(a_m+b_m\right)x^m+\ldots +\left(a_0+b_0\right) |
|---|---|
| Numerical example | \left(2.7x^5-1.8x^3+0.9x-2\right)+\left(3.8x^4+2x^3-x+5.1\right)=\\ 2.7x^5+3.8x^4+0.2x^3-0.1x+3.1 |
By definition, m and n will be non-negative integers, and the coefficients will remain constant. Therefore, the result is another polynomial.
Subtraction will work the same way as addition. Assuming m<n,
| General example | \left(a_{n}x^{n}+\ldots+a_{0}\right)-\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\a_{n}x^n+\ldots +\left(a_m-b_m\right)x^m+\ldots +\left(a_0-b_0\right) |
|---|---|
| Numerical example | \left(\frac{2}{3}x^4+\frac{3}{5}x^2+\frac{1}{2}\right)-\left(\frac{1}{4}x^3+\frac{3}{5}x^2+\frac{4}{7}x+\frac{3}{4}\right)=\\ \frac{2}{3}x^4-\frac{1}{4}x^3-\frac{4}{7}x-\frac{1}{4} |
The coefficients will remain constant, and the exponents will be non-negative integers. The result is another polynomial.
To multiply polynomials, we apply the distributive property which will require us to use the product of powers property of exponents when multiplying variables:
| General example | \left(a_{n}x^{n}+\ldots+a_{0}\right)\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\ \left(a_{n}b_{m}\right)x^{n+m}+\ldots+\left(a_{n}b_0\right)x^n+\ldots+\left(a_{0}b_m\right)x^m+\ldots+\left(a_0b_0\right) |
|---|---|
| Numerical example | \left(5x^2+2x+1\right)\left(x^2-3x+5\right)= 5x^4-13x^3+20x^2+7x+5 |
Because n and m were non-negative integers, n+m will also be non-negative. The exponents will still be constants, so the result is another polynomial.
Examples
Example 1
Determine whether each of the following can be classified as a polynomial.
y=5x^2y+\dfrac{4}{3}xy-2y
\dfrac{4}{x-3}
f(x)=\dfrac{6x^2y}{5}-\sqrt{2}xy+0.5xy^2-4^{-1}
Example 2
Consider the polynomial expression \left(x^2y + 4xy - x^2\right) - \left(5xy^2 - 2xy + 9\right).
Fully simplify the polynomial expression.
State the degree of the simplified polynomial.
Example 3
Form a fully simplified polynomial expression for the perimeter of the rectangle shown.
Example 4
Fully simplify the polynomial expression \left(3 + 2a\right)^3.
Example 5
A rectangular swimming pool is 16\text{ yds} long and 6\text{ yds} wide. It is surrounded by a pebble path of uniform width x\text{ yds}.
Find an expression for the area of the path in terms of x. Fully simplify your answer.
A polynomial is an expression or equation with constant coefficients and non-negative integer exponenets. Polynomials are closed under addition, subtraction, and multiplication which means the sum, difference, or product of polynomials is another polynomial.