How To Find The Leading Term Of A Polynomial

On this post we explicate what is the leading term of a polynomial. Likewise, y'all will run into 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 10 with the highest exponent.

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

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

The highest degree element of the above polynomial is 5x^three (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. And so, following the previous case, the leading coefficient of the polynomial would be 5.

Too, the leading term of a polynomial is used to identify when a polynomial is monic. In the following link you tin meet what a monic polynomial is.

➤ What is a monic polynomial?

Examples of how to detect the leading term of a polynomial

At present that we know how to identify the leading term of a polynomial, we are going to practise with several examples.

Example of the leading term of a polynomial of degree five:

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

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

Example of the leading term of a polynomial of degree half-dozen:

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

The term with the maximum degree of the polynomial is x⁶, then that is the leading term of the polynomial. Call back that if the variable is not accompanied past whatsoever number, information technology ways that the coefficient is 1, consequently, the leading coefficient of this polynomial is i.

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

Example of the leading term of a polynomial of degree ix:

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

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

Case 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 -2x³y^iv, since it is the highest caste monomial of the polynomial.

In this exercise we must pay attending since the caste of a term with 2 variables is not calculated in the aforementioned way equally the caste of a term with simply one variable.