Example Question #1 : Zeroes Of Polynomials
The graph of a function is shown below, with labels on the y-axis hidden.
Determine which of the following functions best fits the graph above.
Use the zeroes of the graph to determine the matching function. Zeroes are values of x where . In other words, they are points on the graph where the curve touches zero.
Visually, you can see that the curve crosses the x-axis when , , and . Therefore, you need to look for a function that will equal zero at these x values.
A function with a factor of will equal zero when , because the factor of will equal zero. The matching factors for the other two zeroes, and , are and , respectively.
The answer choice has all of these factors, but it is not the answer because it has an additional zero that would be visible on the graph. Notice it has a factor of , which results in a zero at . This additional zero that isn't present in the graph indicates that this cannot be matching function.
is the answer because it has all of the required factors and, as a result, the required zeroes, while not having additional zeroes. Notice that the constant coefficient of negative 2 does not affect where the zeroes are.
Example Question #2 : Zeroes Of Polynomials
Which of the functions below best matches the graphed function?
First, look at the zeroes of the graph. Zeroes are where the function touches the x-axis (i.e. values of where ).
The graph shows the function touching the x-axis when , , and at a value in between 1.5 and 2.
Notice all of the possible answers are already factored. Therefore, look for one with a factor of (which will make when ), a factor of to make when , and a factor which will make when is at a value between 1.5 and 2.
This function fills the criteria; it has an and an factor. Additionally, the third factor, , will result in when , which fits the image. It also does not have any extra zeroes that would contradict the graph.