Polynomial Standard Form
Introduction
A polynomial is an expression consisting of variables and constants which can have two or more terms. A polynomial expression includes operations of addition, subtraction, multiplication, division and/or exponents of non-negative integers.A polynomial is used to represent a mathematical relationship between certain variables.
What Do You Mean by Polynomial in Standard Form?
A standard form polynomial is a polynomial where the terms are arranged in order from the highest degree to the lowest. The degree of the polynomial is crucial for writing it in standard form since terms are written in decreasing order of the power of `x`. When a polynomial is written in its standard form, the term with the largest variable power appears first, followed by the following terms in decreasing order of variable power. In polynomials expressed in standard form, the leading term is the first term, and the leading coefficient is the coefficient that comes before it.
A polynomial in standard form is expressed as follows: `f(x) = a_nx^n + a_(n-1)x^(n-1) + a_(n-2)x^(n-2) +... + a_1x + a_0`.
According to the definition of polynomials in standard form, the exponents of the polynomials must be expressed in decreasing order. An example of a polynomial written in standard form can be `7x^9 + 5x^7 - 9x^3 +11x - 2`. As you can see, the exponents in this polynomial are arranged in descending order of power. The standard form of polynomials is used to simplify calculations.
Degree of a Polynomial
The degree of a polynomial is a fundamental concept in algebra, providing insight into its behavior and characteristics. In simple terms, the degree of a polynomial is the highest power of the variable in any of its terms.
For example, consider the polynomial ` 3x^2 - 2x + 5 `. The highest power of the variable ` x ` is ` x^2 `, so the degree of this polynomial is `2`.
For a multi-variable polynomial, how to find the degree of a polynomial? . It follows the same strategy except that the degree of a multi-variable polynomial is the highest sum of the powers of a term in the polynomial. For example, consider the polynomial `7xy^5 + 4x^5y^3 - 5x^3y^2 + 20x`.
The degree of the first term `(7xy^5)` is `1 + 5 = 6`
The degree of the second term `(4x^5y^3)` is `5 + 3 = 8`
The degree of the third term `(- 5x^3y^2)` is `3 + 2 = 5`
The degree of the fourth term `(20x)` is `1`
The highest exponent is `8` and term with the highest exponent is `4x^5y^3`.
Hence `4x^5y^3` is the leading coefficient with the polynomial in standard form being `4x^5y^3 + 7xy^5- 5x^3y^2 + 20x`.
Polynomials can have different degrees, and the degree influences various aspects of their behavior. Here are some key points about polynomial degrees:
`1`. Degree Determines Behavior: The degree of a polynomial impacts its end behavior. For instance, polynomials with odd degrees (like ` x^3 ` or ` x^5 `) will have opposite directions at their ends, while those with even degrees (like ` x^2 ` or ` x^4 `) will have the same end behavior.
`2`. Number of Roots: The degree of a polynomial also provides information about the number of roots or solutions it may have. According to the Fundamental Theorem of Algebra, a polynomial of degree ` n ` will have exactly ` n ` roots, counting multiplicity.
`3`. Leading Coefficient: In a polynomial expression, the coefficient of the term with the highest power of the variable (the leading term) is called the leading coefficient. This coefficient can influence the overall behavior of the polynomial, particularly in terms of its growth or decay as the variable's value becomes large.
`4`. Polynomial Classification: Polynomials are often classified based on their degree. A polynomial of degree `0` is a constant polynomial, degree `1` is a linear polynomial, degree `2` is a quadratic polynomial, degree `3` is a cubic polynomial, and so on.
Understanding the degree of a polynomial is crucial in various mathematical contexts, including calculus, algebra, and data analysis. It helps mathematicians and scientists analyze functions, solve equations, and model real-world phenomena accurately.
Steps to Write a Polynomial in Standard Form
To write a polynomial in standard form, you need to arrange its terms in descending order of the degree of the variable. Here are the steps to do that:
`1`. Collect Like Terms: Combine all like terms together. Like terms are those that have the same variable(s) raised to the same power. For example, in the polynomial `5x^2 + 2x - 3x^2 - 4`, the like terms are `5x^2` and `-3x^2`, while `2x` and `-4` are not like terms.
`2`. Arrange Terms in Descending Order of Degree: Once you've collected like terms, arrange them in descending order of the degree of the variable. The term with the highest degree should come first, followed by the term with the next highest degree, and so on. For example, in the polynomial `3x^2 + 2x - 5x^2 - 4`, after combining like terms, you get `5x^2 - 3x^2 + 2x - 4`. Now arrange them in descending order of degree: `2x^2 + 2x - 4`.
`3`. Check for Leading Coefficient: For standard form in polynomials, the leading coefficient (the coefficient of the term with the highest degree) should ideally be positive. If it's negative, you can multiply the entire polynomial by `-1` to make it positive.
After following these steps, you'll have your polynomial written in standard form, with the terms arranged in descending order of degree. For instance, the polynomial `5x^2 + 2x - 3x^2 - 4` would be written in standard form as `2x^2 + 2x - 4`.
Classification of Polynomials
Polynomials can be divided into groups according to their power and degree. We can categorize polynomials based on the number of terms:
Monomial: Polynomial with only one term. For example, `−5xy, 6y^2,` etc.
Binomial: Polynomial with two terms. For example `x + 5, y^2 + 5,` and `3x^3 − 7,` etc.
Trinomial: Polynomial with three terms. For example `3x^3 + 15x − 10, x + y + z, 6x + y^2 − 7,` etc.
In general, any polynomial with more that one term can be called polynomial with “poly” meaning “many”.
The following table shows the classification of some standard form polynomials based on the highest degree and the number of terms.
Adding and Subtracting Polynomials in Standard Form
When we add or subtract polynomials, knowing how to write a polynomial in standard form is very helpful. The proces of adding or subtracting polynomials is very similar to the process of adding or subtracting numbers in standard form. Let’s illustrate it through an example:
Example `1`: Add the polynomials `(x - x^2 + 5)` and `(6x^2 - 10 + 2x)`.
Solution:
Step `1`: Align the terms according to their powers of `x`:
The polynomials in their standard form will be `(-x^2 + x + 5)` and `(6x^2 + 2x - 10)`
Step `2`: Combine like terms:
Add the coefficients of terms with the same exponent.
`(-x^2 + x + 5) + (6x^2 + 2x - 10)`
`= (-x^2 + 6x^2) + (x + 2x) + (5 -10)`
`= 5x^2 + 3x - 5`
Example `2`: Subtract the polynomial `(12 + 3^2 - 5x)` from `(2x + 4x^2 - 10)`.
Solution:
Step `1`: Align the terms according to their powers of `x`:
The polynomials in their standard form will be `(3^2 - 5x + 12)` from `(4x^2 + 2x - 10)`.
Step `2`: Combine like terms:
Pair the like terms together and then subtract
`(4x^2 + 2x - 10) - (3x^2 - 5x + 12)`
`= (4x^2 - 3x^2) + (2x + 5x) + (-10 -12)`
`= x^2 + 7x - 22`
Solved Examples
Example `1`: Write the polynomial ` 2x^3 - 5x^2 + 4x - 7` in standard form.
Solution:
The polynomial is already in standard form because its terms are arranged in descending order of degree.
Example `2`: Perform the operation `(3x^2 - 2x + 5) + (4x^2 + 7x - 3)` and write the result in standard form.
Solution:
\( (3x^2 - 2x + 5) + (4x^2 + 7x - 3) \)
\(= 3x^2 - 2x + 5 + 4x^2 + 7x - 3 \)
\( = (3x^2 + 4x^2) + (-2x + 7x) + (5 - 3) \)
\( = 7x^2 + 5x + 2 \)
The result `7x^2 + 5x + 2` is in standard form.
Example `3`: Perform the operation `(6x^3 + 2x - 3x^2 + 1) - (5x^2 + 2x^3 + 7 - 3x)` and write the result in standard form.
Solution:
\( (6x^3 + 2x - 3x^2 + 1) - (5x^2 + 2x^3 + 7 - 3x) \)
\( = (6x^3 - 3x^2 + 2x + 1) - (2x^3 + 5x^2 - 3x + 7) \)
\( = 6x^3 - 3x^2 + 2x + 1 - 2x^3 - 5x^2 + 3x - 7 \)
\( = (6x^3 - 2x^3) + (-3x^2 - 5x^2) + (2x + 3x) + (1 - 7) \)
\( = 4x^3 - 8x^2 + 5x - 6 \)
The result `4x^3 - 8x^2 + 5x - 6` is in standard form.
Example `4`: Classify the polynomial `2x^4 - 3x^2 + 5x - 1` by its degree.
Solution:
The degree of the polynomial is the highest power of the variable, which is `4`. Therefore, the polynomial is a quartic polynomial.
Example `5`: Classify the polynomial `4x^2 - 2x + 7` by its degree.
Solution:
The degree of the polynomial is the highest power of the variable, which is `2`. Therefore, the polynomial is a quadratic polynomial.
Practice Problems
Q`1`. Which of the following represents a polynomial in standard form?
- `2x + 5x^3 - 1`
- `5x - 1 + 2x^3`
- `x^2 - 3x + 2`
- `3x + 2x^2 - 5`
Answer: c
Q`2`. What is the result of adding the polynomials `3x^2 + 2x - 1` and `4x^2 - 3x + 2`?
- `7x^4 - x + 1`
- `7x^2 - x + 1`
- `7x^2 - x - 3`
- `7x^2 + x + 1`
Answer: b
Q`3`. Which of the following polynomials is classified as a quadratic polynomial?
- `3x^3 + 2x^2 - x + 1`
- `5x^4 - 2x^3 + 4x^2 - x + 3`
- `2x^2 + 3x - 1`
- `4x + 2`
Answer: c
Q`4`. What is the standard form of the polynomial `4x^3 - 2x^2 + 5x + 3`?
- `4x^3 + 5x - 2x^2 + 3`
- `4x^3 - 2x^2 + 5x + 3`
- `4x^3 - 2x + 5x^2 + 3`
- `4x^2 - 2x^3 + 5x + 3`
Answer: b
Q`5`. What is the result of subtracting the polynomial `3x^2 - 2x + 1` from `5x^2 + 4x - 2`?
- `2x^2 + 6x - 3`
- `2x^2 + 6x - 1`
- `2x^2 - 6x + 3`
- `2x^2 - 6x + 1`
Answer: a
Frequently Asked Questions
Q`1`. What is the standard form of a polynomial?
Answer: The standard form of a polynomial is when the terms are arranged in descending order of the degree of the variable, with like terms combined. For example, `3x^2 + 2x - 5x^2 - 4` can be written in standard form as `-2x^2 + 2x - 4`.
Q`2`. How do you add or subtract polynomials?
Answer: To add or subtract polynomials, you simply combine like terms. Add or subtract the coefficients of like terms while keeping the variable(s) and their exponents the same. For example, to add `2x^2 + 3x - 5` and `x^2 - 2x + 7`, you add the coefficients of like terms: `2x^2 + x^2 = 3x^2`, `3x - 2x = x`, and `-5 + 7 = 2`, giving you `3x^2 + x + 2`.
Q`3`. What are the classifications of polynomials based on their degrees?
Answer: Polynomials are classified based on their degrees. A polynomial of degree `0` is a constant polynomial, degree `1` is a linear polynomial, degree `2` is a quadratic polynomial, degree `3` is a cubic polynomial, and so on. A polynomial of degree `n` is often referred to as an `n`-degree polynomial or simply an `n`-th degree polynomial.
Q`4`. Why is it important to arrange polynomials in standard form?
Answer: Arranging polynomials in standard form makes it easier to identify the highest degree term, leading coefficient, and overall structure of the polynomial. It also facilitates operations like addition, subtraction, multiplication, and division, as it ensures like terms are aligned for straightforward computation.
Q`5`. How do you classify a polynomial based on its degree and number of terms?
Answer: Polynomials can also be classified based on their degrees and number of terms. For example, a polynomial with one term is called a monomial (e.g., `3x^2`), two terms is called a binomial (e.g., `2x - 1`), and three terms is called a trinomial (e.g., `x^2 + 3x - 5`). The degree of the polynomial is determined by the highest power of the variable(s) present among its terms.