Leading term of a polynomial

On this post we explain what is the leading term of a polynomial. Also, you will see several examples on finding the leading term of a polynomial.

What is the leading term of a polynomial?

The definition of the leading term of a polynomial is as follows:

The leading term of a polynomial is the term with the highest degree of the polynomial, that is, the leading term of a polynomial is the term that has the x with the highest exponent.

For example, the leading term of the following polynomial is 5x3:

5x^3-4x^2+2x+1

The highest degree element of the above polynomial is 5x3 (monomial of degree 3), therefore that is the leading term of the polynomial.

On the other hand, the coefficient of the leading term is called the leading coefficient of a polynomial. So, following the previous example, the leading coefficient of the polynomial would be 5.

Also, the leading term of a polynomial is used to identify when a polynomial is monic. In the following link you can see what a monic polynomial is.

What is a monic polynomial?

Examples of how to find the leading term of a polynomial

Now that we know how to identify the leading term of a polynomial, we are going to practice with several examples.

  • Example of the leading term of a polynomial of degree 5:

7x^2+2x^5+x^3

The leading term of the polynomial is 2x5 because it is the term with the highest power of x.

  • Example of the leading term of a polynomial of degree 6:

x^6-5x^3+2x-10

The term with the maximum degree of the polynomial is x6, so that is the leading term of the polynomial. Remember that if the variable is not accompanied by any number, it means that the coefficient is 1, consequently, the leading coefficient of this polynomial is 1.

Note that if the polynomial is in standard form, the leading term is the first term in the polynomial.

  • Example of the leading term of a polynomial of degree 9:

-3x^9+7x^6+5x^4+3x^2+x

The term of the polynomial whose exponent is the highest is -3x9, so the leading term of the polynomial is -3x9. Note that the negative sign is also part of the leading term.

  • Example of the leading term of a polynomial with two variables:

3x^5+x^3y-2x^3y^4+10y^6

The leading term of the polynomial is -2x3y4, since it is the highest degree monomial of the polynomial.

In this exercise we must pay attention since the degree of a term with two variables is not calculated in the same way as the degree of a term with only one variable.

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