What is a Polynomial?
A polynomial is an algebraic expression that has more than one term consisting of variables and coefficients. In a polynomial, the variables can only have non-negative integer exponents, and the coefficients can be any real number. Polynomials are widely used in mathematics, especially in algebra and calculus, to represent various mathematical functions.
For example, the following are some examples of polynomial expressions:
3x^2 + 2x + 1
4y^3 – 3y^2 + 2y – 1/2
5z^4 + 6z^2 – 7z
Writing Polynomials in Standard Form
Standard form of a polynomial expression is when the terms are arranged in descending order of their exponents, and the coefficients are written in a particular format. The format requires the first coefficient to be positive and the coefficients of any missing term to be zero. Writing polynomials in standard form is a crucial concept in mathematics, and this article aims to guide you on how to do it correctly.
The first step in writing a polynomial expression in standard form is to make sure that all the terms are in descending order. For example, let’s consider the polynomial expression below:
5x^3 + 2x^2 – 3x + 1
To write this expression in standard form, we start by arranging the terms in descending order of their exponents:
5x^3 + 2x^2 – 3x + 1 = 5x^3 + 2x^2 – 3x^1 + 1x^0
Next, we rearrange the terms in a specific format ensuring the first coefficient is positive and including zero coefficients for any missing terms.
5x^3 + 2x^2 – 3x + 1 = 5x^3 + 2x^2 + 0x^1 – 3x^0 + 1x^0
This polynomial is now in standard form.
If we consider the polynomial expression:
2x^2 – 1
We notice that there is no “x^1” term, which means we need to include a zero-coefficient for that particular term:
2x^2 + 0x^1 – 1x^0
Finally, we rearrange the terms in standard form:
2x^2 + 0x^1 – 1x^0 = 2x^2 + 0x^1 – 1
It is important to note that it’s not usually required to include the “+0” term, so we abbreviate our final expression to:
2x^2 – 1
Writing polynomial expressions in standard form is vital in mathematics as it helps to make the expressions more comfortable to understand, analyze, and solve. To ensure that the polynomial is in standard form, make sure all terms are arranged in decreasing order of their exponents, have positive first coefficients, and include zero coefficients for any missing terms. Practice implementing these steps when writing polynomial expressions in standard form, and you will improve your mathematics skills.
The Meaning of Standard Form
Polynomials are mathematical expressions which consist of terms that are separated by addition or subtraction signs. They can include various variables and constants, and their powers and coefficients can change. Writing polynomials in standard form is a useful and common practice among mathematicians and students alike. It allows us to easily identify the highest degree of the polynomial and the leading coefficient. In this article, we will explain what standard form is, why it is important, and how to write polynomials in standard form.
How to Write Polynomials in Standard Form
Writing polynomials in standard form requires the rearrangement of the terms so that they are in decreasing order of exponents. This means that the term with the highest degree is listed first, followed by the next highest degree, and so on. For example, the polynomial:
4x3 + 2x2 – 5x + 3
would be written in standard form as:
4x3 + 2x2 – 5x + 3
Notice how the terms are now organized with the highest degree term first, followed by the next highest degree, and so on. This makes it easy to determine the degree of the polynomial (which is 3), and the leading coefficient (which is 4).
It’s important to note that when writing polynomials in standard form, we also want to ensure that all the like terms are combined. That means that terms with the same variable and exponent are grouped together. For example, the polynomial:
3x2 + 5x – 2 + 2x2 – 4x
Can be simplified and written in standard form as:
5x2 + x – 2
When there are negative terms in the polynomial, it’s also important to remember that we want the coefficients to be positive. This can be achieved through multiplication by -1. For example, the polynomial:
-2x3 – 4x2 + 8x
Can be simplified and written in standard form as:
-2x3 – 4x2 + 8x
Multiplying each term by -1:
2x3 + 4x2 – 8x
Now we can rearrange the terms to put them in decreasing order of exponents:
2x3 + 4x2 – 8x
Once all the terms are in decreasing order of exponents, and all like terms are combined, the polynomial is in standard form.
Writing polynomials in standard form is a simple but important practice in mathematics. It allows us to quickly and easily identify the highest degree of the polynomial, the leading coefficient, and all like terms. By following the steps outlined in this article, you can write any polynomial in standard form and be confident in your mathematical abilities.
Collect Like Terms
Before we dive into writing polynomials in standard form, it’s important to understand the concept of “like terms.” Like terms refers to terms that have the same variables raised to the same power. For example, 2x and 5x are like terms because they both have the variable x raised to the power of 1. However, 2x and 5x^2 are not like terms because they have different powers of x.
The first step to writing a polynomial in standard form is to identify and combine like terms. Let’s look at an example polynomial:
3x^2 + 2x^3 – 5x + 4x^2 – 2
To collect like terms, we’ll focus on the variables and their exponents. In this polynomial, we have two terms with x^2 and one term with x, which are like terms. We can combine these terms to simplify the polynomial:
2x^3 + 7x^2 – 5x – 2
Now that we’ve simplified the polynomial by collecting like terms, we can move on to writing it in standard form.
Arrange Terms in Descending Order
Writing polynomials in standard form requires arranging the terms in descending order of exponents. The term with the highest exponent comes first, followed by the terms with the next highest exponents, and so on.
For example, let’s take the polynomial:
3x^2 + 4x^3 – 2x + 7
First, we must arrange the terms in descending order of exponents:
4x^3 + 3x^2 – 2x + 7
As we can see, the term with the highest exponent is 4x^3, followed by 3x^2, and so on.
It’s important to note that when arranging terms in descending order, it’s not only the exponents that matter but also the coefficients. For example, let’s take the polynomial:
5x^3 – 3x + 2x^2 + 7
Again, we must arrange the terms in descending order of exponents:
5x^3 + 2x^2 – 3x + 7
Notice that although the term 2x^2 has a lower exponent than 5x^3, it comes before it in the polynomial because its coefficient is higher.
Arranging terms in descending order may seem simple, but it’s a crucial step in writing polynomials in standard form.
What Is Standard Form For Polynomials?
Polynomials are expressions that contain variables raised to a power and accompanied by coefficients, which are constants that multiply the variables. You can write polynomials in many different forms, but one of the most useful is standard form.
Standard form for polynomial means writing the terms with descending order of exponents on the variable. In other words, the variable with the highest exponent is written first and the one with the lowest written last. For example, the standard form for the polynomial 2x^3 + 4x^2 – 3x + 1 is 2x^3 + 4x^2 – 3x + 1.
Writing polynomials in standard form can help you simplify the expression, identify the degree of the polynomial, and compare two polynomials to determine which one has a higher degree or what common factors they share.
Steps For Writing Polynomials In Standard Form
To write a polynomial in standard form, follow these steps:
Step 1: Arrange the terms in descending order by the exponent of the variable.
For example, suppose you are given the polynomial P(x) = 3x^4 – 2x^3 – 5x^2 + 7x – 1, to put the terms in order from highest power to lowest, you start by identifying the term with the highest power on the variable, which is the one with the exponent of 4. So, that should be the first term in the rewritten expression. The terms with exponents of 3, 2, 1, and 0 follow in that order until you have listed all the terms.
Step 2: Combine like terms (terms with the same exponent on the variable).
For example, after arranging the terms of P(x) in descending order, we get 3x^4 – 2x^3 – 5x^2 + 7x – 1. To write it in standard form, we need to combine the terms that have the same exponent. We can see that there are no other terms with exponent 4 in this polynomial except 3x^4, so we just rewrite that. The term with exponent 3 is -2x^3, and there is no other term with that exponent, so we just copy it down. The term with exponent 2 is -5x^2, we have to check whether there are any other terms with exponent 2 so we can combine them. There are no such terms, so we rewrite that. The term with exponent 1 is 7x, and there is no other term with that exponent, so we just copy it down. Finally, the constant term is -1 and there are no other constant terms, so we rewrite that.
Step 3: Rewrite the polynomial with the terms in descending order (highest power first) and combined like terms.
For the polynomial P(x) = 3x^4 – 2x^3 – 5x^2 + 7x – 1, we can group the like terms, making it easy to see what we need to rearrange and rewrite. To do this, we get rid of any parentheses and combine any like terms. So, we will get: 3x^4 + (-2)x^3 + (-5)x^2 + 7x + (-1) which is equivalent as 3x^4 – 2x^3 – 5x^2 + 7x – 1. That’s the standard form of the polynomial!
3 Common Mistakes To Avoid When Writing Polynomials In Standard Form
When writing polynomials in standard form, there are some common mistakes that you should avoid:
Mistake #1: Writing the terms in ascending order.
Always begin by writing the term that has the highest power first and work your way down to the constant term. This will ensure that your polynomial is in standard form.
Mistake #2: Confusing the degree of the polynomial with the leading coefficient.
The degree of a polynomial is the highest power of the variable, whereas the leading coefficient is the coefficient of the term with the highest degree. Make sure you know which one you are looking for when writing a polynomial in standard form.
Mistake #3: Failing to combine like terms.
When writing a polynomial in standard form, it is important to combine any like terms before rearranging them in descending order. This will ensure that your polynomial is fully simplified and in its most compact form.
Examples for Practice
Practice makes perfect! Here are some more examples of polynomials that you can practice writing in standard form:
1. Q(x) = 2x^3 + 3x + 4x^2 – 2
2. R(x) = x^2 + 2x^3 – 3x + 1
3. S(x) = 4x – 7x^3 + 2x^2 + 5
To put the polynomials in standard form, follow the steps discussed earlier:
Q(x) = 2x^3 + 4x^2 + 3x – 2 (Standard form)
R(x) = 2x^3 + x^2 – 3x + 1 (Standard form)
S(x) = -7x^3 + 2x^2 + 4x + 5 (Standard form)
Now try working on more examples on your own and check your work against a solution key for practice!