Question

For f(x)=3x(x^2+1)(x+3)^2, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept.

Answer 1

i can help but would like us to walk through it together so that you learn: can you take a first stab at it? What would be the first easy-to-spot zero of the function f(x)?

Answer 2

Well for the 3x, the zero would probably be zero.

Answer 3

correct!

Answer 4

And for the (x+3)^2, it would be -3.

Answer 5

perfect, and that one qould have multiplicity of 2 because of the square, like (x+3)(x+3)

Answer 6

now the middle part is a term x^2+1 so the question is when is that term 0?

Answer 7

That's correct. But to balance out that section wouldn't x^2 have to equal -1?

Answer 8

yes, it would, and that is why they mention "real" zeros - as opposed to "imaginary" ones. The equation \[x^2=-1\] has a solution but only in imaginary numbers, namely it is the number \[i\].

Answer 9

so that is an imaginary zero and is not part of the solution they are asking for

Answer 10

So are the zeros 0, i, and -3?

Answer 11

yes 0 (multipl. 1), -3 (multipl. 2) would be the answer to the problem. and yes "i" is another zero (but not real)

Answer 12

do you know what to do about the second part of the question?

Answer 13

It's when the multiplicities are even or odd, correct?

Answer 14

Even bounces and odd crosses?

Answer 15

yes you got it

Answer 16

Thanks again.

Answer 17

yw!

Answer 18

Refer to the attached plot.

Answer 19

Thanks!

Answer 20

very nice @robtobey