How to find the leading coefficient of a polynomial

Q: What is the leading coefficient and degree of a polynomial?

We call the term containing the highest power of x (i.e. anxn) the leading term, and we call an the leading coefficient. The degree of the polynomial is the power of x in the leading term. We have already seen degree 0, 1, and 2 polynomials which were the constant, linear, and quadratic functions, respectively.

In a polynomial, the leading term is the term with the highest power of \(x\). For example, the leading term of \(7+x-3x^2\) is \(-3x^2\).

Table of Contents Show

Examples of how to find the leading coefficient of a polynomial
Leading coefficients and graphs
What is the leading coefficient?
How do you find the degree and coefficient of a polynomial?
What is the leading coefficient and degree of a polynomial?

The leading coefficient of a polynomial is the coefficient of the leading term. In the above example, the leading coefficient is \(-3\).

The graph of the polynomial function f ( x ) = a n x n + a n − 1 x n − 1 + ... + a 1 x + a 0 eventually rises or falls depends on the leading coefficient ( a n ) and the degree of the polynomial function.

Case

End Behavior of graph

When n is odd and a n is positive Graph falls to the left and rises to the right
When n is odd and a n is negative Graph rises to the left and falls to the right
When n is even and a n is positive Graph rises to the left and right
When n is even and a n is negative Graph falls to the left and right

Example:

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function f ( x ) = − x 3 + 5 x .

Solution:

Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure.

The leading coefficient is #13#
The degree is #4#

Explanation:

To obtain the leading coefficient it is necessary to rewrite the equation in canonical form, that is with the terms listed in descending order of their #x# exponents:

#F(x) = 13x^4-14x^3-8x^2-6x+14#

The leading coefficient is the coefficient of the highest order #x# term

The degree is the degree (exponent) of the highest order #x# term

The degree is the sum of the exponents on all terms. Our exponents are #5, 2# and #1#, which sum up to #8#. This is the degree of our polynomial #g(x)#.

The leading term of a polynomial is just the term with the highest degree, and we see this is #3x^5#.

The leading coefficient is just the number multiplying the highest degree term. The coefficient on #3x^5# is #3#.

The constant term is just a term without a variable. In our case, the constant is #1#.

For end behavior, we want to consider what our function goes to as #x# approaches positive and negative infinity.

In our polynomial #g(x)#, the term with the highest degree is what will dominate the end behavior. So let's take the limit of it:

#color(blue)(lim_(xrarroo) 3x^5=oo)#

#color(blue)(lim_(xrarr-oo) 3x^5=-oo)#

Our limits describe our limit behavior.

Hope this helps!

Step 1

The degree of a polynomial is the highest degree of its terms.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

The largest exponent is the degree of the polynomial.

Step 2

The leading term in a polynomial is the term with the highest degree.

Step 3

The leading coefficient of a polynomial is the coefficient of the leading term.

The leading term in a polynomial is the term with the highest degree.

The leading coefficient in a polynomial is the coefficient of the leading term.

Step 4

List the results.

Polynomial Degree:

Leading Term:

Leading Coefficient:

On this post we explain what the leading coefficient of a polynomial is and how to find it. Also, you will see several examples on how to identify the leading coefficient of a polynomial.

What is the leading coefficient of a polynomial?
Examples of how to find the leading coefficient of a polynomial
Leading coefficients and graphs

The definition of leading coefficient of a polynomial is as follows:

In mathematics, the leading coefficient of a polynomial is the coefficient of the term with the highest degree of the polynomial, that is, the leading coefficient of a polynomial is the number that is in front of the x with the highest exponent.

For example, the leading coefficient of the following polynomial is 5:

The highest degree term of the above polynomial is 5x3 (monomial of degree 3), therefore the coefficient of the maximum degree term is 5. And, consequently, the leading coefficient of the polynomial is equal to 5.

Note that if a polynomial is in standard form, the leading coefficient will always be the coefficient of the first term.

Moreover, the term with the highest degree is also called leading term. Thus, the leading coefficient is the coefficient of the leading term of the polynomial.

As you can see, to determine the leading coefficient of a polynomial you must know how to calculate the degree of all the terms of a polynomial. When the polynomial has only one variable is quite easy, but finding the leading coefficient when the polynomial has two or more variables it is more complicated. You can see how to calculate the degree of a term with two variables in the following link:

➤ See: degree of a polynomial with two variables

Examples of how to find the leading coefficient of a polynomial

Once we know how to identify the leading coefficient of a polynomial, let’s practice with several solved examples.

Example of the leading coefficient of a polynomial of degree 4:

The highest degree term of the polynomial is 3x 4, so the leading coefficient of the polynomial is 3.

Example of the leading coefficient of a polynomial of degree 5:

The term with the maximum degree of the polynomial is 8x5, therefore, the leading coefficient of the polynomial is 8.

Example of the leading coefficient of a polynomial of degree 7:

The highest degree element of the polynomial is -6x 7, thus, the leading coefficient of the polynomial is -6. Note that the negative sign is also part of the coefficient.

Leading coefficients and graphs

The graph of a polynomial function depends on the sign of the leading coefficient and the exponent of the leading term as follows:

If the leading coefficient is positive and the exponent of the leading term is odd, the graph falls to the left and rises to the right.
If the leading coefficient is negative and the exponent of the leading term is odd, the graph rises to the left and falls to the right.
If the leading coefficient is positive and the exponent of the leading term is even, the graph rises to the left and right.
If the leading coefficient is negative and the exponent of the leading term is even, the graph falls to the left and right.

What is the leading coefficient?

The leading term is the term containing the highest power of the variable: the term with the highest degree. The leading coefficient is the coefficient of the leading term. Because of the definition of the “leading” term we often rearrange polynomials so that the powers are descending: f(x)=anxn+an−1xn−1… a2x2+a1x+a0.

How do you find the degree and coefficient of a polynomial?

Identify the exponents on the variables in each term, and add them together to find the degree of each term. The largest exponent is the degree of the polynomial. The leading term in a polynomial is the term with the highest degree. The leading coefficient of a polynomial is the coefficient of the leading term.

What is the leading coefficient and degree of a polynomial?

We call the term containing the highest power of x (i.e. a_nxⁿ) the leading term, and we call a_n the leading coefficient. The degree of the polynomial is the power of x in the leading term. We have already seen degree 0, 1, and 2 polynomials which were the constant, linear, and quadratic functions, respectively.