Finding and Using the Degree of a Polynomial
Description/Explanation/Highlights
Video Description
This video explains how to find the degree of a polynomial and how to use it to determine the type of polynomial.
Steps and Key Points to Remember
To find the degree of a polynomial, follow these steps:
- Remember a polynomial is made up of one or more terms separated by “+” or “-” signs, that have no variables with negative exponents, fractional exponents, or variables in the denominator.
- In general, to find the degree of a polynomial, find the degree of each term in the polynomial and use the largest degree as the degree of the entire polynomial.
- To find the degree of a term, add up all the exponents on the variables in the term.
- If there are no variables, the degree of a term is zero.
- To find the degree of \(4x^3+3x^2+2x+4\), find the degree of \(4x^3\), \(3x^2\), \(2x\), and \(4\). The degree of \(4x^3\) is 3 since the exponent on the only variable is 3. The degree of \(3x^2\) is 2. Again, the exponent of the only variable is 2. The degree of \(2x\) is 1 since a variable without an exponent is assumed to have an exponent of 1. The degree of \(5\) is 0 since there is no variable. The largest degree of any term is 3, therefore the degree of the polynomial is 3.
- To find the degree of the monomial \(3a^4b^5c^3\) add up all the exponents: 4+5+3=12. Since there is only one term in a monomial, the degree of this term (12) is also the degree of the polynomial.
- In a polynomial with multiple terms and multiple variables such as \(3x^2y+4xy^4+5y^6\) use the same pattern of finding the degree of each term by adding up the exponents and using the largest degree as the degree of the polynomial. In this example the degree of term 1 is 3 by adding the exponent of 2 for x to the exponent of 1 for y. The degree of the second term is 5 (1+4) and the degree of the third term is 6. Therefore, the degree of the polynomial is 6 because 6 is the largest degree of any term.
- Once we determine the degree of a polynomial with a single variable, we can classify the polynomials based on the degree. A polynomial of degree 0 is a constant, degree 1 is linear, degree 2 is a quadratic, degree 3 is a cubic, degree 4 is a quartic, degree 5 is quintic, and although they keep going, anything above 3 is often just called an \(n^{th}\) degree polynomial where \(n\) is the degree of the polynomial.
Here are some key points to keep in mind when finding and using degrees of polynomials.
- Always start by identifying the terms of the polynomial and finding the degree of each.
- Count only the exponents on the variables of a term.
- Add up all exponents on variables in a term to determine its degree.
- The highest degree from all the terms is the degree of the polynomial.
- Use the degree of the polynomial to classify the type of polynomial such as constant (0), linear (1), quadratic (2), cubic (3), quartic (4), quintic (5), etc.
- Polynomials with higher degrees (above 3) are often just classified as an \(n^{th}\) degree polynomial with \(n\) being the degree of the polynomial.
Video Highlights
- 00:00 Introduction
- 00:07 Review definition of a polynomial
- 00:22 Finding the degree of a polynomial
- 01:14 \(3a^4b^5c^3\) example of finding the degree of a monomial with multiple variables
- 01:58 \(3x^2y+4xy^4+5x^6\) example of finding the degree of a trinomial with multiple variables.
- 03:15 Classifying polynomials by degree
- 04:10 Conclusion
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