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Polynomials can be simplified by using the distributive property to distribute the term on the outside of the parentheses by multiplying it by everything inside the parentheses.

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The Degree of a Polynomial is the largest of the degrees of the individual terms.

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We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial.

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Classifying Polynomials | Developmental Math Emporium

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Polynomials come in many forms. They can vary by how many terms, or monomials, make up the polynomial and they also can vary by the degrees of the monomials in the polynomial. In this section, we will look at different ways that we classify polynomials. First, we will classify polynomials by the num…

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It will be a polynomial if it doesn't contain, no square roots of variables, no fractional or negative power on the variables, and no variables in the denominators of any fractions.

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The terms are written in decreasing order of the power of x.

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Liz bought 2 apples, 4 sticks of butter, and a cupcake and spent $35 in all. She made a quick note in the form of an equation: 2a + 4b + c = 35. Do you know that she wrote a polynomial? Observe the image which shows a polynomial and its various parts.

Read on to know more about Standard form polynomial and check out the interactive examples and the interesting practice questions at the end of the page.

Lesson Plan

What Is Meant by Standard Form Polynomial?
A mathematical expression of one or more algebraic terms in which the variables involved have only non-negative integer powers is called a polynomial.
The terms have variables, constants, and exponents.

The standard form polynomial of degree 'n' is:

Examples: \(x^2+8x-9\), \(t^3-5t^2+8\)

Standard Form Polynomial Degree

The degree of a polynomial is the value of the largest exponent in the polynomial.

However, it differs in the case of a single-variable polynomial and a multi-variable polynomial.

​In a single-variable polynomial, the degree of a polynomial is the highest power of the variable in the polynomial.

Observe the terms, the coefficients, the variables, and the constant labeled in the figure.

The highest exponent in the polynomial \(8x^2-5x+6\) is 2 and the term with the highest exponent is \(8x^2\).

In a multi-variable polynomial, the degree of a polynomial is the sum of the powers of the polynomial.

Consider the polynomial p(x)=\(5 x^4y-2x^3y^3+ 8x^2y^3 -12\)

The highest exponent is 6, and the term with the highest exponent is \(2x^3y^3\).

Therefore, the Deg( p(x) ) = 6

The degree of this polynomial \(5 x^4y-2x^3y^3+ 8x^2y^3 -12\) is the value of the highest exponent, which is 6

Standard Form Polynomial Definition

The Standard form polynomial definition states that the polynomials need to be written with the exponents in the decreasing order.

Polynomials are written in the standard form to make calculations easier.

A polynomial is said to be in its standard form, if it is expressed in such a way that the term with the highest degree is placed first, followed by the term which has the next highest degree, and so on.

For example: \(14 x^4-5x^3-11x^2-11x+8\)

You can observe that in this standard form of a polynomial, the exponents are placed in descending order.

These algebraic equations are called the polynomial equations.

How to Write a Polynomial in Standard Form?
The like terms are grouped, added, or subtracted and rearranged with the exponents of the terms in descending order.
\(8x^5+11x^3-6x^5-8x^2\)

=\(8x^5-6x^5+11x^3-8x^2\)

=\(2 x^5+ 11 x^3- 8 x^2\)

Here the polynomial's highest degree is 5 and that becomes the exponent with the first term.

Here are the steps to write any polynomial in the standard form:

Write the terms.

Group all the like terms.

Find the exponent.

Write the term with the highest exponent first.

Write the rest of the terms with lower exponents in descending order.

Write the constant term (a number with no variable) in the end.

Based on the standard polynomial degree, there are different types of polynomials.

\(\therefore\) the standard form is \(4v^8+8v^5-v^3+ 8v^2\)

Find the degree of the monomial: -4t.

The variable is t and its power is 1

Thus, the exponent of this term is 1

The degree of this monomial -4t is 1

Write \(x^4y^2+10 x+5x^3y^5\) in the standard form.

Consider each term and find its degree.

The degree of the term \(x^4y^2\) = 4 + 2=6

The degree of the term 10 x = 1

The degree of the term \(5x^3y^5\) = 3 + 5=8

Arranging the exponents in the descending powers, we get

\(5x^3y^5+ x^4y^2+10 x\) in the standard form.

\(\therefore\) the standard form is \(5x^3y^5+ x^4y^2+10 x\)

Let's Summarize

We hope you enjoyed learning about standard form polynomial with the solved examples and interactive questions. Now, you will be able to easily solve problems related to standard form polynomial.

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1. What is the standard form of cubic polynomial?

2. What is the standard form of a quadratic polynomial with real coefficients?

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Liz bought 2 apples, 4 sticks of butter, and a cupcake and spent $35 in all. She made a quick note in the form of an equation: 2a + 4b + c = 35. Do you know that she wrote a polynomial? Observe the image which shows a polynomial and its various parts.