Polynomials are expressions that include more than one term. These terms can be numbers or variables. For example, the term 9×2 + 8x is a polynomial consisting of two terms, 9×2 and + 8x.

The first term, 9×2, is called the leading term or the *highest degree term*. The **second term**, + 8x, is called the *trailing term* or the **lowest degree term**. The coefficient of the leading term is also the degree of that polynomial.

So in this case, the degree of this polynomial is 2. Polynomials can either be linear or quadratic depending on the degrees of the terms. Linear polynomials only have degrees 1 and 0, where quadratic polynomials have degrees 2 and 1.

## The number of terms

When comparing polynomials, the first step is to look at the degrees of each polynomial. The degree of a polynomial is how many roots or values the polynomial has.

For example, the polynomial x2 + 2x + 1 has *degree two* because it can be solved by the roots x = 0 and x = 1.

The **term count** of a polynomial depends on its degree. A linear (1st degree) polynomial has **one term per variable**, a quadratic (2nd degree) has **two terms per variable**, and so on.

Then, compare the largest term of one polynomial with the smallest term of the other polynomial. If one does not have any terms in common then compare the greatest factor of one with the smallest factor of the other.

## How are the terms arranged?

The order or arrangement of the terms in a polynomial determines what kind of function it is. A polynomial can be a **linear term**, *quadratic term*, or **cubic term depending** on how the terms are arranged.

A linear polynomial is one where the coefficients of the variables are constant. For example, 2x + 4y is a linear polynomial because the coefficients of the variables are 2 and 1, and they do not change when we solve for them.

A quadratic polynomial is one where the highest coefficient of any of the variables is 2. For example, x2 + 3x + 5 is a quadratic polynomial because the highest coefficient of any variable is 2.

A cubic polynomial is one where the highest coefficient of any variable is 3. For example, x3 + **y3 − z3** is a cubic polynomial because the highest coefficient of any variable is 3.

## Example of subtracting polynomials

Polynomials can be subtracted just *like linear functions* can. The difference is that the coefficients and variables change. When subtracting polynomials, the larger coefficient variable is what changes.

To subtract polynomials, you **must first find** the common denominator between the two monomials. In this example, the common denominator is 8. Then, you would subtract the second term of one polynomial from the first term of the other polynomial and then add the opposite of that result in the second polynomial.

This process is easy to remember because you are simply switching the letters of the variables and coefficients. You are **also simply switching** which variable is greater or lesser than another variable.

You can *also subtract ungraded polynomials* but must keep in mind how many zeroes are in each coefficient.

## When doing algebra, it is important to know how to work with polynomials

Polynomials are mathematical expressions that contain more than one term. These terms can be constants, variables, or summations and divisions of variables.

Like all algebraic expressions, polynomials can be written in different forms. The form in which you see a polynomial does not necessarily represent its structure. For example, the polynomial 2×2 + **3x − 1** can be written as 2×2 + (3x − 1), (2×2 + 3x) − 1, or 2×2+(3x−1)−1.

When working with polynomials, it is important to know how to differentiate and integrate them. To differentiate a polynomial, you must first divide it by the leading term of the expression-the term that is multiplied the most times by other terms in the expression. Then, you must subtract the constant term of the expression from the rest of the terms to get just a variable.

## What is a polynomial?

A polynomial is a finite expression that consists of the sum of multiple terms, each consisting of a constant and a variable raised to a constant.

For example, the term 4×2 is a polynomial consisting of the sum of 4x and 2, where x is raised to the power 2.

The term (3x + 1)2 is also a polynomial, consisting of the sum of 3x and 1 squared. Here, both 3x and 1 are constants, and the latter is raised to the power 2.

There are many uses for polynomials, but *one common use* is *solving linear equations using linear algebra*. To do this, you **must first rewrite** your equation as a *linear equation using coefficients* from polynomials.

## How do you tell the difference between two polynomials?

Polynomials are algebraic expressions that consist of constants and multiples of constants and variables. They can also consist of negative constants and variables.

There are two ways to differentiate between two polynomials. The first way is to examine the coefficients of the variables. The second way is to look at the degree of the polynomials.

Examining the coefficients of the *variables means looking* at how many times each variable occurs as a coefficient. For example, if **one variable occurred** as a coefficient twice, then that variable is in the polynomial twice.

The degree of a polynomial refers to how many constant and/or variable terms it has. For example, 9×2 + 8x is a polynomial of degree 2, because it has **two constant terms** and **one variable term**.

## What is the difference between the two polynomials? (9×2 + 8x)-(2×2 + 3x)

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There are two main differences between the two polynomials. The first is how many variables are in the equation. The second is the degree of the polynomial, which is how many levels or steps it takes to get to the top.

The first difference is easy to explain. A linear equation has one variable, a quadratic has two variables, a cubic has three variables, and a quartic has four variables.

The second difference requires more explanation. The degree of a polynomial refers to how many times the variable changes value before reaching the top. For example, in 9×2 + 8x-2×2 + 3x, **x changes twice** before reaching the top, so this polynomial is of degree two.

In general, linear equations have degree one, quadratic equations have degree two, cubic equations have degree three, and quartic equations have degree four.

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