How to Find the Degree of a Polynomial

Step-by-Step Guide

1. 1

   Step 1: Combine like terms.

   Combine all of the like terms in the expression so you can simplify it, if they are not combined already.
   
   Let's say you're working with the following expression: 3x2
   - 3x4
   - 5 + 2x + 2x2
   - x.
   
   Just combine all of the x2, x, and constant terms of the expression to get 5x2
   - 3x4
   - 5 + x.
2. 2

   Step 2: Drop all of the constants and coefficients.

   The constant terms are all of the terms that are not attached to a variable, such as 3 or
   5.
   
   The coefficients are the terms that are attached to the variable.
   
   When you're looking for the degree of a polynomial, you can either just actively ignore these terms or cross them off.
   
   For instance, the coefficient of the term 5x2 would be
   5.
   
   The degree is independent of the coefficients, so you don't need them.
   
   Working with the equation 5x2
   - 3x4
   - 5 + x, you would drop the constants and coefficients to get x2
   - x4 + x. , This is also called putting the polynomial in standard form..
   
   The term with the highest exponent should be first, and the term with the lowest exponent should be last.
   
   This will help you see which term has the exponent with the largest value.
   
   In the previous example, you would be left with
   -x4 + x2 + x. , The power is simply number in the exponent.
   
   In the example,
   -x4 + x2 + x, the power of the first term is
   4.
   
   Since you've arranged the polynomial to put the largest exponent first, that will be where you will find the largest term. , You can just write that the degree of the polynomial = 4, or you can write the answer in a more appropriate form: deg (3x2
   - 3x4
   - 5 + 2x + 2x2
   - x) =
   3.
   
   You're all done., If your polynomial is only a constant, such as 15 or 55, then the degree of that polynomial is really zero.
   
   You can think of the constant term as being attached to a variable to the degree of 0, which is really
   1.
   
   For example, if you have the constant 15, you can think of it as 15x0, which is really 15 x 1, or
   15.
   
   This proves that the degree of a constant is
   0. , Finding the degree of a polynomial with multiple variables is only a little bit trickier than finding the degree of a polynomial with one variable.
   
   Let's say you're working with the following expression: x5y3z + 2xy3 + 4x2yz2 , Just add up the degrees of the variables in each of the terms; it does not matter that they are different variables.
   
   Remember that the degree of a variable without a written degree, such as x or y, is just one.
   
   Here's how you do it for all three terms:x5y3z = 5 + 3 + 1 = 9 2xy3 = 1 + 3 = 4 4x2yz2 = 2 + 1 + 2 = 5 , The largest degree of these three terms is 9, the value of the added degree values of the first term. , 9 is the degree of the entire polynomial.
   
   You can write the final answer like this: deg (x5y3z + 2xy3 + 4x2yz2) =
   9. , Let's say you're working with the following expression: (x2 + 1)/(6x
   -2)., You won't need the coefficients or constant terms to find the degree of a polynomial with fractions.
   
   So, eliminate the 1 from the numerator and the 6 and
   -2 from the denominator.
   
   You're left with x2/x. , The degree of the variable in the numerator is 2 and the degree of the variable in the denominator is
   1.
   
   So, subtract 1 from
   2. 2-1 =
   1. , The degree of this rational expression is
   1.
   
   You can write it like this: deg =
   1.
3. 3

   Step 3: Put the terms in decreasing order of their exponents.

4. 4

   Step 4: Find the power of the largest term.

5. 5

   Step 5: Identify this number as the degree of the polynomial.

6. 6

   Step 6: Know that the degree of a constant is zero.

7. 7

   Step 7: Write the expression.

8. 8

   Step 8: Add the degree of variables in each term.

9. 9

   Step 9: Identify the largest degree of these terms.

10. 10

    Step 10: Identify this number as the degree of the polynomial.

11. 11

    Step 11: Write down the expression.

12. 12

    Step 12: Eliminate all coefficients and constants.

13. 13

    Step 13: Subtract the degree of the variable in the denominator from the degree of the variable in the numerator.

14. 14

    Step 14: Write the result as your answer.

Detailed Guide

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