Video: Degree and Leading Coefficient of One-Variable Polynomials

Consider the polynomial function 𝑓(𝑥) = −8𝑥⁵ + 3𝑥⁴ − 12𝑥⁶ + 5𝑥² − 12. What is its degree? What is its leading coefficient?

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Video Transcript

Consider the polynomial function 𝑓 of 𝑥 is equal to negative eight 𝑥 to the five plus three 𝑥 to the four minus 12𝑥 to the six plus five 𝑥 squared minus 12. What is its degree? And what is its leading coefficient?

The degree of a one-variable polynomial is the highest power or exponent of that variable. The powers or exponents in this polynomial function are five, four, six, and two. We couldn’t factor at the final term, the constant term, as negative 12 multiplied by 𝑥 to the zero, as 𝑥 to the power of zero is equal to one.

The highest power of 𝑥 — it appears in this polynomial function — is six. And so this is the degree of the polynomial. Now let’s consider the leading coefficient. The leading coefficient is the number in front of the variable with the highest power.

We’ve already established that the term with the highest power is the term with the 𝑥 to the six. And the coefficient here is negative 12. Note that the sign of this number is important. The coefficient isn’t 12. It’s negative 12.

It is perhaps more usual to see polynomial functions written in order of decreasing powers of 𝑥. If this were the case, then the leading coefficient is in fact the first coefficient in the polynomial function. However, as we’ve seen in this question, this isn’t always the case. So we need to look carefully through the polynomial in order to find the term with the highest power. The degree of this polynomial function is six. And the leading coefficient is negative 12.

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