Question

1. Given the following polynomial, find
a. the zeros and the multiplicity of each
b. where the graph crosses or touches the x‐axis
c. number of turning points
d. end behavior
f(x) = - x^2 (x+4)^3 (x^2-1)

Answer 1

To find the zeros, deal with each factor separately. Set each factor equal to zero and solve.
\[-x^2=0\]thus\[x=0 \]is a factor. It occurs twice and so has a multiplicity of 2. Do this for (x+4)^3 and (x^2-1) to get the other zeros. The multiplicity is just the exponent on the factor.

Answer 2

eseidl your first response didnt show up. Something and something is a factor???

Answer 3

To find the zeros, deal with each factor separately. Set each factor equal to zero and solve.
−x2=0
thus
x=0
is a factor. It occurs twice and so has a multiplicity of 2. Do this for (x+4)^3 and (x^2-1) to get the other zeros. The multiplicity is just the exponent on the factor.

Answer 4

show up now?

Answer 5

yes I see

Answer 6

should say -x^2 not -x2...the copy and paste missed that

Answer 7

b. the graph touches the x-axis at the roots of the polynomial with an even multiplicity. If the root has an odd multiplicity, the graph will cross the x-axis at that root.

Answer 8

So the graph will touch the x-axis at x=0 because the multiplicity of the root (=2) is even.

Answer 9

good luck :)

Answer 10

thanks

Answer 11

how do I find the end behavior ?

Answer 12

You have to logic out what the graph will do as x approaches -infinity and as x approaches +infinity. We look at the largest power of x to see this because it has the largest effect as x gets very large (positive or negative). If you expand your polynomial, the largest term will be -x^7. In this case, when x gets very large (and is negative) the -x^7 term dominates the value that y takes. A very large negative number to the power of 7 is a VERY large negative number. But we are dealing with -x^7, not x^7 so we will get a VERY large positive number as gets gets bigger and bigger in the negative x direction. This corresponds to region II in the cartesian plane (x is negative, y is positive). If x is very large positive number the x^7 will be VERY large as well. But again we have -x^7 so we get a VERY large negative value for y as x increases in the positive x direction. Again this corresponds to region IV (x is positive, y is negative). The dotted line indicates the "rest of the function" and is not meant to be accurate.