Polynomials in standard form can also be referred to as the standard form of a polynomial which means writing a polynomial in the descending order of the power of the variable. To write a polynomial in a standard form, the degree of the polynomial is important as in the standard form of a polynomial, the terms are written in decreasing order of the power of x. The standard form of polynomial is given by, f(x) = anx n + an-1x n-1 + an-2x n-2 + . + a1x + a0, where x is the variable and ai are coefficients.
In this article, we will learn how to write the standard form of a polynomial with steps and various forms of polynomials. Read on to know more about polynomial in standard form and solve a few examples to understand the concept better.
| 1. | What is Standard Form of Polynomial? |
| 2. | Meaning of Polynomial in Standard Form |
| 3. | Standard Form of Polynomial Degree |
| 4. | Steps for Writing Standard Form of Polynomial |
| 5. | Addition and Subtraction of Standard Form of Polynomial |
| 6. | FAQs on Standard Form of Polynomial |
The standard form of a polynomial is a way of writing a polynomial such that the term with the highest power of the variables comes first followed by the other terms in decreasing order of the power of the variable. The first term in the standard form of polynomial is called the leading term and its coefficient is called the leading coefficient. The standard form of a polynomial is given by, f(x) = anx n + an-1x n-1 + an-2x n-2 + . + a1x + a0.
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: anx n + an-1x n-1 + an-2x n-2 + . + a1x + a0. For example, x 2 + 8x - 9, t 3 - 5t 2 + 8.
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 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 of power. These algebraic equations are called polynomial equations.
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. The highest exponent in the polynomial 8x 2 - 5x + 6 is 2 and the term with the highest exponent is 8x 2 . So, the degree is 2.
In a multi-variable polynomial, the degree of a polynomial is the highest sum of the powers of a term in the polynomial. Consider the polynomial p(x) = 5 x 4 y - 2x 3 y 3 + 8x 2 y 3 -12.
| Term | Sum of the powers | Degree |
|---|---|---|
| 5 x 4 y | 4+1 | 5 |
| 2x 3 y 3 | 3+3 | 6 |
| 8x 2 y 3 | 2+3 | 5 |
| 12 | 0 | 0 |
The highest exponent is 6, and the term with the highest exponent is 2x 3 y 3 . Therefore, the Deg p(x) = 6. The degree of this polynomial 5 x 4 y - 2x 3 y 3 + 8x 2 y 3 -12 is the value of the highest exponent, which is 6.
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: 8x 5 + 11x 3 - 6x 5 - 8x 2 = 8x 5 - 6x 5 + 11x 3 - 8x 2 = 2x 5 + 11x 3 - 8x 2 . 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:
Based on the standard polynomial degree, there are different types of polynomials.
| Polynomial | Degree | Standard Form | Example |
|---|---|---|---|
| Constant | 0 | f(x) = c | f(x) = 3 |
| Linear | 1 | f(x) = ax + b | f(x) = 2x + 1 |
| Quadratic | 2 | f(x) = ax 2 + bx + c | f(x) = x 2 + 5x + 6 |
| Cubic | 3 | f(x) = ax 3 + bx 2 + cx + d | f(x) = x 3 + 2x 2 - 5x - 10 |
Polynomials can be categorized based on their degree and their power. Based on the number 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 6y 2 . A binomial is a type of polynomial that has two terms. For example x + 5, y 2 + 5, and 3x 3 − 7. While a Trinomial is a type of polynomial that has three terms. For example 3x 3 + 15x − 10, x + y + z, and 6x + y − 7.
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.
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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 descending order, we get the standard polynomial as 4v 8 + 8v 5 - v 3 + 8v 2 . Answer: 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. Answer: Therefore, the degree is 1.
Example 3: Write x 4 y 2 + 10 x + 5x 3 y 5 in the standard form. Solution Consider each term and find its degree. The degree of the term x 4 y 2 = 4 + 2 = 6 The degree of the term 10x = 1 The degree of the term 5x 3 y 5 = 3 + 5 = 8 Arranging the exponents in the descending powers, we get, Answer: 5x 3 y 5 + x 4 y 2 + 10x in the standard form.
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