Find the degree and the leading coefficient of the polynomial function 𝑓 of 𝑥 equals three 𝑥 to the fourth power plus two 𝑥 cubed plus five 𝑥 squared plus seven.
Let’s recap some of the key terms in this question. We’re told that 𝑓 of 𝑥 is a polynomial function. It’s the sum of monomials. In this case, our polynomial function is a function in just one variable; it’s 𝑥. So what does it mean to find the degree of a polynomial?
Well, the degree of a polynomial is found by looking for the highest exponent of its terms. That’s fairly straightforward when we’re working in one variable. For instance, our term five 𝑥 squared has an exponent of two, whilst our constant term seven, which is technically seven 𝑥 to the zeroth power, has an exponent of zero. If we were dealing with a function that had a number of variables, the exponents of each term is the sum of the exponents of the variables that appear in each term. So to find the degree of 𝑓 of 𝑥, let’s continue identifying the exponent of the terms. Three 𝑥 to the fourth power has an exponent of four, whilst two 𝑥 cubed has an exponent of three. Since the highest exponent of the terms is four, that’s the degree of our polynomial. So with that in mind, how can we find the leading coefficient? Well, luckily, we can use what we’ve already worked out, since the leading coefficient is the coefficient of the term with the highest exponent.
Now, remember, we identified the degree of the polynomial to be four since the term with the highest exponent was three 𝑥 to the fourth power. So what is its coefficient? Well, the coefficient essentially tells us how many 𝑥 to the fourth powers we actually have. In this case, we see that that is three. And so the leading coefficient LC must be equal to three. And so we’ve completed this question. The degree of this polynomial 𝑓 of 𝑥 is four, whilst its leading coefficient is three.