Video Transcript
What is the leading coefficient of the polynomial function 𝑓 of 𝑥 equals two 𝑥 squared plus three 𝑥 cubed plus 𝑥 plus four?
We have a polynomial function defined 𝑓 of 𝑥, where, of course, a polynomial function is the sum of monomials. So, how do we define the leading coefficient of a polynomial function? Well, let’s begin by defining what we mean by the exponent of a term within a polynomial. The exponent of a term is the sum of the exponents of the variables that make up that term. For instance, imagine we had the expression three 𝑥 cubed 𝑦 squared plus four 𝑥𝑦 cubed. The exponent of the term three 𝑥 cubed 𝑦 squared is three plus two, which is five, and the exponent of four 𝑥𝑦 cubed is one plus three, which is four. With that in mind, we can then define the leading coefficient of a polynomial. This is the coefficient of the term with the highest exponent.
So, with all of this in mind, let’s inspect in a little more detail each term of the polynomial function. The first term is two 𝑥 squared. So, the exponent here is two. We notice that our polynomial function, in fact, just contains one variable 𝑥, so we don’t actually need to add any exponents. The exponent of three 𝑥 cubed is three. The exponent of 𝑥, well, that’s 𝑥 to the first power, so it’s one. Similarly, a constant four is the same as four 𝑥 to the zeroth power. So, its exponent is zero. This means the term with the highest exponent then is three 𝑥 cubed.
So, what is the leading coefficient? What is the coefficient of this term? The coefficient tells us how many 𝑥 cubes we have. So here, the coefficient of 𝑥 cubed, and hence the leading coefficient, is three. And so, we found the leading coefficient of the function 𝑓 of 𝑥 equals two 𝑥 squared plus three 𝑥 cubed plus 𝑥 plus four. It’s three.
