Standard Form of Polynomial
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) = anxn + an-1xn-1 + an-2xn-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.
What is 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) = anxn + an-1xn-1 + an-2xn-2 + ... + a1x + a0.
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: anxn + an-1xn-1 + an-2xn-2 + ... + a1x + a0. 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 of power. These algebraic equations are called polynomial equations.
Standard Form of 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. The highest exponent in the polynomial 8x2 - 5x + 6 is 2 and the term with the highest exponent is 8x2. 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 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 For Writing Standard Form of Polynomial
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 term 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 | 0 | f(x) = c | f(x) = 3 |
| Linear | 1 | f(x) = ax + b | f(x) = 2x + 1 |
| Quadratic | 2 | f(x) = ax2 + bx + c | f(x) = x2 + 5x + 6 |
| Cubic | 3 | f(x) = ax3 + bx2 + cx + d | f(x) = x3 + 2x2 - 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 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 Standard Form of Polynomial
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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FAQs on Standard Form of Polynomial
What is the Standard Form of Polynomial?
Standard Form of Polynomial 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. The standard form of polynomial is given by, f(x) = anxn + an-1xn-1 + an-2xn-2 + ... + a1x + a0, where x is the variable and ai are coefficients.
When Can You Say That Given Form is the Standard Form of Polynomial?
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 Form 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.