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Polynomial Functions - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

A polynomial function is a function comprised of more than one power function where the coefficients are assumed to not equal zero. The term with the highest degree of the variable in polynomial functions is called the leading term. All subsequent terms in a polynomial function have exponents that decrease in value by one.

I want to talk about Polynomial functions. Now the definition of a Polynomial function is written on the board here and I want to walk you through it cause it is kind of a little bit theoretical if a polynomial functions is one of the form p of x equals a's of n, x to the n plus a's of n minus 1, x to the n minus 1 plus and so on plus a's of 2x squared plus a of 1x plus a's of x plus a's of 0.
Where the numbers a's of i are constants and n is non negative integer. Now what is a polynomial it's really just a finite sum of power functions. Each of these guys here is a power function right? You've got x to sum integer power times a coefficient some constant.
And there may be so many constants that I need to number them with an index I can't just go a to z assuming that there are 26 of them. So it's always some constant number times x to an integer power. Now the number n you know when we write it in this form with descending power the number n is called the degree of the polynomial that's the highest power of x that you find in the polynomial.
And the term with the highest power of x is called the leading term right. All of these are terms of the polynomial but this is the leading term even if it's not written first. The term with the highest power backs is the leading term. And ace of n the number in front is the leading coefficient. So let's take a look at some examples, I got 5 examples here, f of x equals 12x-5 this is a polynomial right? Constant times x to a power, to the power 1 minus constant times x the power of 0. The leading coefficient here is 12 right the leading term is 12x so the leading coefficient is 12 and the degree, that's the power of the highest power of x is 1 so this is a degree 1 polynomial.
g of x, 2x squared minus 5x+6, this is the leading term, this is the term with a highest power of x, the leading coefficient is 2 and the degree the power of the highest power of x is 2 so quadratic right? Functions that are, polynomial functions with degree 1 or a linear, linear functions and with degree 2 quadratic.
Now what about this one h of x equals 4x cube minus x of the 4. Here the term with the highest power of x is negative x to the fourth, this is the leading term and so the leading coefficient would be minus 1. Don't forget this minus cancels the coefficient, so negative 1 and the degree would be 4. A constant function k of x equals 9, this is still a polynomial because we think of it as 9 times x to the 0, so the leading coefficient would be 9 and if you think of this as 9 times x to the 0 the degree is 0 this is a 0 degree polynomial. All constant functions are 0 degree polynomials.
Now m of x is not written in expanded format it's written in factored form, so we have to figure out what the leading term is going to be. I don't have to multiply this all out I just need to multiply enough to get the leading term which is 3x cubed right? This is going to give me the leading term, the highest power of x. So the leading coefficient will be 3 and the degree will be 3 so that's what we call a cubic polynomial. So remember polynomials are basically just finite sums of power functions right? Constant times x to an integer power and the leading term is really important, the leading terms determines the end behavior of the polynomial as we'll see and so we need to be able to find out the leading coefficient and the degree of the polynomial in order to determine it's shape.

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