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Polynomial in Standard Form: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Polynomial in Standard FormPolynomial in Standard Form: Definition and ExamplesTable of ContentsPolynomial in Standard Form Definition of Polynomial in Standard Form

A polynomial in standard form refers to a polynomial whose terms are arranged in the descending order of the degree of the variables, from highest to lowest. In this form, the highest degree term is placed at the beginning of the polynomial, followed by terms with decreasing exponential values. This organization helps in simplifying and performing various operations on polynomials. The standard form of a polynomial with degree nnn can be written as anxn+an−1xn−1+…+a1x+a0a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{1}x + a_{0}an​xn+an−1​xn−1+…+a1​x+a0​.

The degree of a polynomial in standard form is simply the degree of the first term, also called the leading term. For a polynomial with a single variable, the degree is the highest exponent of that variable. In polynomials with multiple variables, the degree of each term is calculated by finding the sum of the exponents of all variables in that term, and the degree of the polynomial is the highest among these sums. The coefficient of the leading term is called the leading coefficient.

Examples of Polynomial in Standard Form Example 1: Converting a Polynomial to Standard Form Problem:

Convert the polynomial to standard form: −3x2+x4−5x+5x3+1-3x^{2} + x^{4} - 5x + 5x^{3} + 1−3x2+x4−5x+5x3+1.

Step-by-step solution:

Step 1, Identify all terms and their degrees:

Degree of −3x2=2-3x^{2} = 2−3x2=2 Degree of x4=4x^{4} = 4x4=4 Degree of −5x=1-5x = 1−5x=1 Degree of 5x3=35x^{3} = 35x3=3 Degree of 1=01 = 01=0

Step 2, Arrange the terms in descending order of degree (highest to lowest):

x4x^{4}x4 (degree 444) comes first 5x35x^{3}5x3 (degree 333) comes second −3x2-3x^{2}−3x2 (degree 222) comes third −5x-5x−5x (degree 111) comes fourth 111 (degree 000) comes last

Step 3, Write the polynomial in standard form by putting these terms together: x4+5x3−3x2−5x+1x^{4} + 5x^{3} - 3x^{2} - 5x + 1x4+5x3−3x2−5x+1

Example 2: Finding the Degree of a Polynomial with Multiple Variables Problem:

What is the degree of a polynomial 5x3+xy4−3xy2−45x^{3} + xy^{4} - 3xy^{2} - 45x3+xy4−3xy2−4? Write it in the standard form.

Step-by-step solution:

Step 1, Find the degree of each term by adding the exponents of each variable:

Degree of 5x3=35x^{3} = 35x3=3 (since exponent of xxx is 333) Degree of xy4=1+4=5xy^{4} = 1 + 4 = 5xy4=1+4=5 (exponent of xxx is 111, exponent of yyy is 444) Degree of −3xy2=1+2=3-3xy^{2} = 1 + 2 = 3−3xy2=1+2=3 (exponent of xxx is 111, exponent of yyy is 222) Degree of −4=0-4 = 0−4=0 (constant term has degree 000)

Step 2, Identify the term with the highest degree: xy4xy^{4}xy4 has the highest degree of 5, so it should come first in standard form

Step 3, Arrange all terms in descending order of degree:

xy4xy^{4}xy4 (degree 555) comes first 5x35x^{3}5x3 and −3xy2-3xy^{2}−3xy2 (both degree 333) come next −4-4−4 (degree 000) comes last

Step 4, Write the polynomial in standard form: xy4+5x3−3xy2−4xy^{4} + 5x^{3} - 3xy^{2} - 4xy4+5x3−3xy2−4

The degree of the polynomial is 555.

Example 3: Adding Polynomials in Standard Form Problem:

Add the following polynomials: p(x)=3x2+2x3−5p(x) = 3x^{2} + 2x^{3} - 5p(x)=3x2+2x3−5 and q(x)=12x2+3x3−1q(x) = 12x^{2} + 3x^{3} - 1q(x)=12x2+3x3−1.

Step-by-step solution:

Step 1, Rewrite both polynomials in standard form (arranging terms by descending powers):

p(x)=2x3+3x2−5p(x) = 2x^{3} + 3x^{2} - 5p(x)=2x3+3x2−5 q(x)=3x3+12x2−1q(x) = 3x^{3} + 12x^{2} - 1q(x)=3x3+12x2−1

Step 2, Align like terms (terms with the same degree) from both polynomials:

p(x)=2x3+3x2−5p(x) = 2x^{3} + 3x^{2} - 5p(x)=2x3+3x2−5 q(x)=3x3+12x2−1q(x) = 3x^{3} + 12x^{2} - 1q(x)=3x3+12x2−1

Step 3, Add like terms from both polynomials:

For x3x^{3}x3 terms: 2x3+3x3=5x32x^{3} + 3x^{3} = 5x^{3}2x3+3x3=5x3 For x2x^{2}x2 terms: 3x2+12x2=15x23x^{2} + 12x^{2} = 15x^{2}3x2+12x2=15x2 For constant terms: −5+(−1)=−6-5 + (-1) = -6−5+(−1)=−6

Step 4, Write the final sum in standard form: p(x)+q(x)=5x3+15x2−6p(x) + q(x) = 5x^{3} + 15x^{2} - 6p(x)+q(x)=5x3+15x2−6

Comments(2)AAppDeveloperYuriNovember 4, 2025This glossary page on polynomial in standard form is great! It helped my students grasp the concept easily. Clear explanations are a huge plus!

SShepherdLeoNovember 4, 2025I've been struggling to explain polynomial standard form to my students. This page's clear def and examples made it a breeze! Thanks!

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