how to find the degree of a polynomial

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Polynomial means "many terms," and it can refer to a variety of expressions that can include constants, variables, and exponents. For example, x - 2 is a polynomial; so is 25. To find the degree of a polynomial, all you have to do is find the largest exponent in the polynomial.^[1] If you want to find the degree of a polynomial in a variety of situations, just follow these steps.

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
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.
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3

Put the terms in decreasing order of their exponents. This is also called putting the polynomial in standard form. ^[2]. 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.
4

Find the power of the largest term. 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.
5

Identify this number as the degree of the polynomial. 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) = 4. You're all done.^[3]
6

Know that the degree of a constant is zero. 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.

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1
Write the expression. 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
2
Add the degree of variables in each term. 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:^[4]
- deg(x⁵y³z) = 5 + 3 + 1 = 9
- deg(2xy³) = 1 + 3 = 4
- deg(4x²yz²) = 2 + 1 + 2 = 5
3

Identify the largest degree of these terms. The largest degree of these three terms is 9, the value of the added degree values of the first term.
4

Identify this number as the degree of the polynomial. 9 is the degree of the entire polynomial. You can write the final answer like this: deg (x⁵y³z + 2xy³ + 4x²yz²) = 9.

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1

Write down the expression. Let's say you're working with the following expression: (x² + 1)/(6x -2).^[5]
2

Eliminate all coefficients and constants. 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.
3

Subtract the degree of the variable in the denominator from the degree of the variable in the numerator. 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.
4

Write the result as your answer. The degree of this rational expression is 1. You can write it like this: deg [(x² + 1)/(6x -2)] = 1.

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Question

What is the degree of a polynomial?

In the case of a polynomial with only one variable (such as 2x³ + 5x² - 4x +3, where x is the only variable),the degree is the same as the highest exponent appearing in the polynomial (in this case 3). In the case of a polynomial with more than one variable, the degree is found by looking at each monomial within the polynomial, adding together all the exponents within a monomial, and choosing the largest sum of exponents. That sum is the degree of the polynomial. For example, in the expression 2x²y³ + 4xy² - 3xy, the first monomial has an exponent total of 5 (2+3), which is the largest exponent total in the polynomial, so that's the degree of the polynomial.
Question

What is degree of 1/x^4 + x^2?

1 / (x^4) is equivalent to x^(-4). So the highest (most positive) exponent in the polynomial is 2, meaning that 2 is the degree of the polynomial.
Question

What about a polynomial with multiple variables that has one or more negative exponents in it?

Combine the exponents found within a given monomial as you would if all the exponents were positive, but you would subtract the negative exponents.
Question

What is the multi-degree of a polynomial?

The multi-degree of a polynomial is the sum of the degrees of all the variables of any one term.
Question

How do I find the degree of the polynomials and the leading coefficients?

The degree is the same as the highest exponent appearing in the polynomial. Coefficients have a degree of 1.
Question

What is the degree of expression x/y?

When no exponent is shown, you can assume the highest exponent in the expression is 1. Therefore, the degree of this monomial is 1.
Question

How do I find the degree of a polynomial that is (x^2 -2)(x+5)=0?

The degree is the same as the highest exponent appearing in the final product, so you just multiply the two factors and you'll wind up with x³ as one of the terms in the product. That's the highest exponent in the product, so 3 is the degree of the polynomial.
Question

What's the degree of 3x-8?

One. The term 3x is understood to have an exponent of 1.
Question

What is the degree of (5y^3/x^2)+4x?

The degree is 3.
Question

How do I find proper and improper fractions?

A proper fraction is one whose numerator is less than its denominator. An improper fraction is one whose numerator is equal to or greater than its denominator.

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This just shows the steps you would go through in your mind. You don't have to do this on paper, though it might help the first time. If you do it on paper, however, you won't make a mistake.
By convention, the degree of the zero polynomial is generally considered to be negative infinity.
For the third step, linear terms like x can be written as x ¹ and non-zero constant terms like 7 can be written as 7x ⁰

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To find the degree of a polynomial with one variable, combine the like terms in the expression so you can simplify it. Next, drop all of the constants and coefficients from the expression. Then, put the terms in decreasing order of their exponents and find the power of the largest term. The power of the largest term is the degree of the polynomial. To find the degree of a polynomial with multiple variables, write out the expression, then add the degree of variables in each term. The power of the largest term is your answer! If you want to learn how to find the degree of a polynomial in a rational expression, keep reading the article!

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