Standard Form Polynomial

Standard Form Polynomial
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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.

polynomial

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 is called a polynomial.

The terms have variables, constants, and exponents.

The standard form polynomial of degree 'n' is:

                 \(a_n x^n+.......a_1 x +a_0\)

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.

polynomial

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)=\(5 x^4y-2x^3y^3+ 8x^2y^3 -12\)

Term Sum of the powers Degree
\(5 x^4y\) 4+1 5
\(2x^3y^3\) 3+3 6
\(8x^2y^3\) 2+3 5
\(12\) 0 0

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.


 

 

 

tips and tricks
Tips and Tricks
  • Keep the powers of the exponents in the descending order.
  • Keep the constant term in the end.

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.

Explanation

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.

Types of Polynomials

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

Polynomial Constant  Linear Quadratic Cubic
Degree 0 1 2 3
Standard Form c \(ax\)+\(b\) \(ax^2\)+\(bx\)+\(c\) \(ax^3\)+\(bx^2\)+\(cx\)+\(d\)
Example 3 \(2x\)+\(1\) \(x^2\)+\(5x\)+\(6\) \(x^3\)+\(2x^2\)-\(5x\)-\(10\)
 
important notes to remember
Important Notes
  • In a standard form, the terms are ordered from the highest degree to the lowest degree.
  • A variable with no exponent is considered to be raised to the power 1, for example in the polynomial \(2x^2+3x\), 3x has the exponent value as 1.
  • A constant's exponent is always raised to the power 0, which makes it equal to 1. For example, in the polynomial, \(4x+ 6\), 6 is considered as 6 x\(^0\) and forms the last term of the polynomial.

Solved Examples

Example 1

 

 

Write \(8v^2+4v^8+8v^5-v^3\) in the standard form.

Solution

The highest degree of this polynomial is 8 and the corresponding term is \(4v^8\).

The second highest degree is 5 and the corresponding term is \(8v^5\).

Arranging the exponents in the descending order, we get the standard polynomial as \(4v^8+8v^5-v^3+ 8v^2\)

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

 

 

Find the degree of the monomial: -4t.

Solution

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

\(\therefore\) the degree is 1
Example 3

 

 

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

Solution

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 \(5x^3y^5+ x^4y^2+10 x\)

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

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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Frequently Asked Questions (FAQs)

1. What is the standard form of cubic polynomial?

The standard form of a cubic function p(x) = \(ax^3 + bx^2 + cx + d\), where the highest degree of this polynomial is 3. a, b,c are the variables raised to the powers 3, 2 and 1 respectively and d is the constant.

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

The standard form of a quadratic polynomial p (x) = \(ax^2 + bx + c\), where a, b, and c are real numbers, and a≠ 0

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