1.4 Polynomials
Learning Objectives
- Identify the degree and leading coefficient of polynomials.
- Add and subtract polynomials.
- Multiply polynomials.
- Use FOIL to multiply binomials.
- Perform operations with polynomials of several variables.
Degree of a Polynomial
The degree of a polynomial can be quickly determined visually by looking for the highest exponent term of the polynomial. If a term is a composed of several different variables you add the exponents of each variable. Let’s look at the table below to see several examples.
Example 1.4.1
Polynomial |
Type |
Degree |
|---|---|---|
| 9 | monomial | 0 |
| [latex]3xy^3[/latex] | monomial | 4 |
| [latex]4a^5+15ab^2+2[/latex] | trinomial | 5 |
| [latex]t+8[/latex] | binomial | 1 |
| [latex]r^7+4r^3+3r^2+r+6[/latex] | polynomial | 7 |
| [latex]5+ab+a^2by[/latex] | trinomial | 4 |
Adding and Subtracting Polynomials
To add and subtract polynomials focus on simplifying the like terms in each polynomial. Recall that like terms have the exact same variable part. For instance, the terms [latex]8xy^2[/latex]and [latex]3xy^2[/latex] are like terms because the both have [latex]xy^2[/latex]as the variable part.
Example 1.4.2
First add the given polynomials then subtract :
- [latex](7x^2-4x+5 ) \& (x^2-7x+3)[/latex]
- [latex](14y^2+6y-4) \& (3y^2+8y+5)[/latex]
Solution:
First we perform the addition operation by adding the coefficients of the like terms and leaving the variable parts unchanged.
- [latex](7x^2-4x+5) + (x^2-7x+3)[/latex] = [latex]8x^2-11x+8[/latex]
- [latex](14y^2+6y-4) + (3y^2+8y+5)= 17y^2+14y+1[/latex]
Next for the subtraction process, we subtract the coefficients of like terms being very careful of any sign changes.
- [latex](7x^2-4x+5) - (x^2-7x+3)[/latex] = [latex]6x^2+3x+2[/latex]
- [latex](14y^2+6y-4) - (3y^2+8y+5)= 11y^2-2y-9[/latex]
