Write the Polynomial in Standard Form. Then Name the Polynomial Based on Its Degree 8-4x^2+10x^2+2x

Polynomial In Standard Form

Polynomials in standard form can also be referred to as the standard form of a polynomial which means writing a polynomial in the descending power of the variable. To write a polynomial in a standard form, the degree of the polynomial is important as the standard form of a polynomial, the terms are written in decreasing order of the power of x. Read on to know more about polynomial in standard form and solve a few examples to understand the concept better.

1.	Meaning of Polynomial in Standard Form
2.	Standard Form in Polynomial Degree
3.	Steps to Writing a Polynomial in Standard Form
4.	Addition and Subtraction of Polynomials in Standard Form
5.	FAQs on Polynomial in Standard Form

Meaning of Polynomial in Standard Form

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: \(a_n x^n+.......a_1 x +a_0\). For example, x² + 8x - 9, t³ - 5t² + 8.

Definition of Polynomial in Standard Form

The Standard form polynomial definition states that the polynomials need to be written with the exponents in 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 x⁴ - 5x³ - 11x² - 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 polynomial equations.

Standard Form in 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 8x² - 5x + 6 is 2 and the term with the highest exponent is 8x².

Polynomial in Standard Form

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 x⁴y - 2x³y³ + 8x²y³ -12.

Term	Sum of the powers	Degree
5 x⁴y	4+1	5
2x³y³	3+3	6
8x²y³	2+3	5
12	0	0

The highest exponent is 6, and the term with the highest exponent is 2x³y³. Therefore, the Deg p(x) = 6. The degree of this polynomial 5 x⁴y - 2x³y³ + 8x²y³ -12 is the value of the highest exponent, which is 6.

Steps to Writing a Polynomial in Standard Form

Writing a polynomial in standard form is done depending on the degree as we saw in the previous section. The like terms are grouped, added, or subtracted and rearranged with the exponents of the terms in descending order. For example: 8x⁵ + 11x³ - 6x⁵ - 8x² = 8x⁵ - 6x⁵ + 11x³ - 8x² = 2x⁵ + 11x³ - 8x². Here the polynomial's highest degree is 5 and that becomes the exponent with the first term. Let us look at the steps to writing the polynomials in standard form:

Step 1: Write the terms.
Step 2: Group all the like terms.
Step 3: Find the exponent.
Step 4: Write the term with the highest exponent first.
Step 5: Write the rest of the terms with lower exponents in descending order.
Step 6: Write the constant term (a number with no variable) in the end.

Types of Polynomial

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

Polynomial	Degree	Standard Form	Example
Constant or Zero	0	c	3
Linear	1	ax + b	2x + 1
Quadratic	2	ax² + bx + c	x² + 5x + 6
Cubic	3	ax³ + bx² + cx + d	x³ + 2x² - 5x - 10

Polynomials can be categorized based on their degree and their power. Based on the numbers of terms, there are mainly three types of polynomials that are: Monomials is a type of polynomial with a single term. For example: x, −5xy, and 6y². A binomial is a type of polynomial that has two terms. For example x + 5, y² + 5, and 3x³− 7. While a Trinomial is a type of polynomial that has three terms. For example 3x³ + 15x − 10, x + y + z, and 6x + y − 7.

Addition and Subtraction of Polynomials in Standard Form

Addition and subtraction of polynomials are two basic operations that we use to increase or decrease the value of polynomials. Whether you wish to add numbers together or you wish to add polynomials, the basic rules remain the same. The only difference is that when you are adding 34 to 127, you align the appropriate place values and carry the operation out. However, when dealing with the addition and subtraction of polynomials, one needs to pair up like terms and then add them up. Otherwise, all the rules of addition and subtraction from numbers translate over to polynomials. Have a look at the image given here in order to understand how to add or subtract any two polynomials.

Addition and Subtraction of Polynomials

Example on Polynomial in Standard Form

Example 1: Write 8v² + 4v⁸ + 8v⁵ - v³ in the standard form.

Solution

The highest degree of this polynomial is 8 and the corresponding term is 4v⁸.

The second highest degree is 5 and the corresponding term is 8v⁵.

Arranging the exponents in descending order, we get the standard polynomial as 4v⁸ + 8v⁵ - v³ + 8v².

Therefore, the standard form is 4v⁸ + 8v⁵ - v³ + 8v².
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 x⁴y² + 10 x + 5x³y⁵ in the standard form.

Solution

Consider each term and find its degree.

The degree of the term x⁴y² = 4 + 2 = 6

The degree of the term 10x = 1

The degree of the term 5x³y⁵ = 3 + 5 = 8

Arranging the exponents in the descending powers, we get,

5x³y⁵+ x⁴y² +10x in the standard form.

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Practice Questions on Polynomials in Standard Form

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FAQs on Polynomial in Standard Form

What is the Meaning of Polynomial in Standard Form?

Polynomial in standard form means writing the polynomials with the exponents in decreasing order to make the calculation easier. The standard form of a polynomial is expressed by writing the highest degree of terms first then the next degree and so on.

When Can You Say That Given Polynomial is in Standard Form?

A polynomial is said to be in standard form when the terms in an expression are arranged from the highest degree to the lowest degree. The standard form helps in determining the degree of a polynomial easily.

What are the Steps to Writing a Polynomial in Standard Form?

The steps to writing the polynomials in standard form are:

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.

What are the Types of Polynomials in Standard From Based on Degree?

Based on the degree, the polynomial in standard form is of 4 types:

Zero or Constant Polynomial
Linear Polynomial
Quadratic Polynomial
Cubic Polynomial

What is the Standard Form of Cubic Polynomial?

The standard form of a cubic function p(x) = ax³ + bx² + cx + d, where the highest degree of this polynomial is 3. a, b, and c are the variables raised to the power 3, 2, and 1 respectively and d is the constant.

What is the Standard Form of a Quadratic Polynomial with Real Coefficients?

The standard form of a quadratic polynomial p(x) = ax² + bx + c, where a, b, and c are real numbers, and a ≠ 0.

Is Zero a Polynomial?

Number 0 is a special polynomial called Constant Polynomial.

How to Write Standard Form in 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. In a multi-variable polynomial, the degree of a polynomial is the sum of the powers of the polynomial.

Write the Polynomial in Standard Form. Then Name the Polynomial Based on Its Degree 8-4x^2+10x^2+2x

Source: https://www.cuemath.com/algebra/standard-form-polynomial/