The question is: Determine if ƒ(x) = 3x2 – 2x3 – x – 9 is a polynomial function. State the degree and leading coefficient.

Question

I want to better understand polynomials but our teacher just tells us to figure it out on our own and i'm lost.

Answer 1

Hey Lindsey,

I know how rough it can be when a teacher just leaves you to figure things out on your own, so I hope I can help!

To find the degree of a polynomial look at the exponents on the variable, ask yourself, "What is the largest power (degree) that a variable is raised to?"

Example:

What is the degree of the following polynomial function?

f(x)=2x³-x⁴-x+2

1. First look at all of the "x"s, our variable, and find the one that has the largest exponent (they may not always be in order from largest to smallest).
2. -x⁴ has the highest degree (the largest exponent).
3. The highest exponent/degree is 4 which means that the degree of the polynomial is 4.

A coefficient is the big number in front of a variable, this is what we call a constant (a number that stays the same even when the variable changes). Terms are the groups of numbers or numbers and variables when they are being multiplied together.

Example:

For 2x³:

1. The coefficient is 2
2. The degree is 3
3. The term is 2x³

To find the leading coefficient we need to rearrange our polynomial so that the terms are in order from largest exponent to smallest. Then we look at the first term, this is called our leading term. The big number (coefficient) in front of the variable of the first term is our leading coefficient.

Example:

What is the leading coefficient of the following polynomial?

f(x)=2x³-x⁴-x+2

1. Rearrange the polynomial in order from largest exponent to smallest f(x)= -x⁴+2x³-x+2
2. Pick out the leading (first) term -x⁴
3. Pick out the coefficient of the leading term -1

So your leading coefficient in this example would be -1

Answer 2

Polynomial? Yes, it consists of several monomials joined by addition and subtraction.

Degree? 3, that is the greatest of all the exponents

Leading coefficient? -2, that is the coefficient of the monomial with the highest degree.