How to Calculate Degree of Polynomial

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Polynomials are a fundamental concept in mathematics, extensively used in algebra and calculus. They represent mathematical expressions composed of variables and coefficients, often involving addition, subtraction, multiplication, and exponentiation. One crucial aspect when dealing with polynomials is determining their degree – a measure of the highest power of the variable in the expression. Understanding how to calculate the degree of a polynomial is essential for various mathematical operations, such as simplifying expressions, factoring, and solving equations. In this article, we will explore the steps and techniques required to accurately calculate the degree of a polynomial, providing readers with a solid foundation in this fundamental aspect of algebra.

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Polynomial means “many terms”, and can be used to refer to a range of expressions that include constants, variables, and exponents. For example, “x-2” is a polynomial; So does “25”. To determine the degree of a polynomial, all you need to do is find the largest exponent in that polynomial. ^{[1] X Research Source} To find the degree of a polynomial in various situations, follow these steps.

Table of Contents

Steps

Polynomial containing At most One Variable

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Combine elements. In case the expression is verbose and collapsible, combine similar terms in the expression. Suppose you are considering the following expression: 3x ² – 3x ⁴ – 5 + 2x + 2x ² – x. Combine all the terms containing x ² , x , and the constants to get a reduced expression like this: 5x ² – 3x ⁴ – 5 + x.

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Ignore constants and coefficients. Ignore all constants that are not attached to the variable, for example, 3 or 5. The coefficients are the numbers associated with the variable. When you want to find the degree of a polynomial, you can omit the constants and coefficients or cross them out. For example, the coefficient of the term 5x ² is 5. The degree of the polynomial does not depend on the coefficients, so you don’t need to worry about them.

• With the expression 5x ² – 3x ⁴ – 5 + x, you remove the constant and the coefficient gets x ² – x ⁴ + x.

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Arrange the remaining terms in descending order of the exponent. Also known as returning the expression to the standard form . ^{[2] X Research Source} The term with the highest exponent is first and the term with the lowest exponent is last. This step will help you determine which term has the largest exponent. In the previous example, we got
-x ⁴ + x ² + x.

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Find the power of the largest term. The power is the value of the exponent. In the -x ⁴ + x ² + x example, the power of the first term is 4. Since the expression is sorted in descending order of the exponent, you can easily determine the term. biggest.

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The value found in the previous step is the degree of the polynomial. You can write the degree of the polynomial = 4, or you can write the answer in full: deg (3x ² – 3x ⁴ – 5 + 2x + 2x ² – x) = 3. That’s it! ^{[3] X Research Sources}

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When you know the degree of a constant is 0. If your polynomial is just a constant, like 15 or 55, the degree of that polynomial is 0. You can think of constants tied to variables of degree 0. i.e. a variable with a value of 1. For example, if you had a constant of 15, you could think of the number as 15x ⁰ , which is in fact 15 x1, and shortened to 15. This worked Prove that the degree of a constant is 0.

A Polynomial Containing Many Variables

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Write expressions. Finding the degree of a polynomial containing many variables is only slightly more complicated than finding the degree of a polynomial containing one variable. Let’s take the following expression as an example:

• x ⁵ y ³ z + 2xy ³ + 4x ² yz ²

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Add the degrees of the variables in each term. You simply add up the degrees of all the weighted variables, whether they are the same or different. Remember that for variables with no particular degree, such as x and y, their degree is 1. So, for the three terms in the above example, we have: ^{[4] X Source rescue}

• x ⁵ y ³ z = 5 + 3 + 1 = 9
• 2xy ³ = 1 + 3 = 4
• 4x ² yz ² = 2 + 1 + 2 = 5

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Determine the largest order. The largest degree between the terms is 9, this is the value when adding the degrees of the elements of the first term.

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This is the degree of the polynomial. 9 is the degree of the whole polynomial. You can write the final result like this: deg (x ⁵ y ³ z + 2xy ³ + 4x ² yz ² ) = 9 .

Expressions As Fractions

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Write expressions. Take the following expression as an example: (x ² + 1)/(6x -2). ^{[5] X Research Sources}

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Remove all coefficients and constants. You do not need coefficients or constants when finding the degree of a polynomial containing fractions. Therefore, removing 1 from the numerator, 6 and -2 from the denominator, we get: x ² /x.

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Subtract the degree of the variable in the numerator from the degree of the variable in the denominator. The degree of the variable in the numerator is 2 and the degree of the variable in the denominator is 1, take 2 minus 1, we have: 2-1 = 1.

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

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Advice

• The instructions above outline the steps you need to take. You don’t have to do all the steps on paper, but it’s better to put them on paper the first time you do these steps because it’s hard to make mistakes when you do them on paper.
• By convention, polynomials with no degree are negative infinity.
• In step 3, terms like x can be written as x ¹ and non-zero constants like 7 can be written as 7 x ⁰

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X

wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 30 people, some of whom are anonymous, have edited and improved the article over time.

This article has been viewed 155,885 times.

Polynomial means “many terms”, and can be used to refer to a range of expressions that include constants, variables, and exponents. For example, “x-2” is a polynomial; So does “25”. To determine the degree of a polynomial, all you need to do is find the largest exponent in that polynomial. ^{[1] X Research Source} To find the degree of a polynomial in various situations, follow these steps.

In conclusion, calculating the degree of a polynomial is a straightforward process that involves determining the highest power of the variable in the polynomial. By examining the terms of the polynomial and identifying the power of the variable in each term, one can easily determine the degree. It is important to correctly identify the degree of a polynomial as it allows for a better understanding of the polynomial’s behavior and properties. Additionally, knowing the degree of a polynomial is crucial for performing operations such as addition, subtraction, multiplication, and division of polynomials. By following the steps outlined in this article, anyone can easily calculate the degree of a polynomial and use this information to further analyze and manipulate polynomials in various mathematical contexts.

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