In this lesson, Professor John Zhu gives an introduction to polynomial functions. He goes over the 3 types of functions which are: no domain restrictions, no discontinuities, and the degree test. He also explains the leading coefficient test when it has a positive or negative leading coefficient, with even or odd degrees.
Polynomial degree test: f(x) changes direction n-1 times
“+” leading coefficient and “even” degree polynomial: both ends up
“+” leading coefficient and “odd” degree polynomial: left end down, right end up
“-” leading coefficient and “even” degree polynomial: both ends down
“-” leading coefficient and “odd” degree polynomial: left end up, right end down
Find the degree of the following polynomial f(x) = 5x7 + x6 + x3 + x
When the function only involves one variable, the degree is the number of the biggest exponent. In this case, the degree is 7
Find the degree of the following polynomial g(x) = (−3847x50 + 123x48 + 456x46 + 6786x44)2(482x2 + 452x12)x2
Again, we're only interested in the largest exponent. Instead of multiplying everything out, we only have to worry about the biggest exponents. When multiplying the same variable, the exponents are added. So degree = 50 + 50 + 12 + 2 =
Using f(x) and g(x) from the previous problems, which is greater, f(99999999999) or g(−99999999999)
No need to actually plug in to compare. We can look at the exponent and leading coefficient. f(x) has a positive leading coefficient, and an odd degree. For very large positive numbers, f(x) is very large and positive. As for g(x), it has a negative leading coefficient and an even degree. A negative number raised to an even exponent is a positive number. A very large positive number multiplied by a negative number is a negative number
f(99999999999) > 0
g(−99999999999) < 0
f(99999999999) > g(−99999999999)
Describe the behavior of the following function at the extremes of x f(x) = −x7 + 200x4 − 50
Negative leading coefficient, odd degree. As x gets very large, the 7 exponent dominates the other parts of the function. When looking at 'edge' behavior, only the largest exponent, or degree, really matters. As x goes to positive infinity, f(x) goes to negative infinity. As x goes to negative infinity, f(x) goes to positive infinity.
Name the degree and type of polynomials of the following functions: f(x) = 3x2 + x
Degree is 2 and it is a quadratic function
Name the degree and type of polynomials of the following functions: f(x) = 5x + 1
Degree is 1 and it is a linear function
Name the degree and type of polynomials of the following functions: f(x) = 2
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.