f(x)=3x^5-10x^4-6x^3+24x^2+11x-6
Zeros: 2, 3, -1 and 1/3
It is a degree 5 polynomial. Why is there only 4 zeros?

Question

f(x)=3x^5-10x^4-6x^3+24x^2+11x-6
Zeros: 2, 3, -1 and 1/3
It is a degree 5 polynomial. Why is there only 4 zeros?

jim_thompson5910 · Answer

If you have listed all the roots, then there must be a root that repeats itself.

jim_thompson5910 · Answer

Example:

y = x^2 + 6x + 9

has 2 roots, but they are the same number: that root is -3 (and it repeats twice)

anonymous · Answer

So how do I find out which one repeats?

jim_thompson5910 · Answer

how did you find those zeros in the first place?

anonymous · Answer

Listed all the possibilities by the Rational Root Theorem, than graphed it. I saw that 2, 3 and -1 was a zero for sure.  So I divided the poly with 2, than 3 and than -1. I was left with a quadratic function which I solved and got 1/3 as answer.

jim_thompson5910 · Answer

do you see any roots where it touches the x-axis, but doesn't cross x-axis?

anonymous · Answer

yes at -1. it touches and than goes back where it came from. would that be a multiplicity?

jim_thompson5910 · Answer

that would imply that there is a double root there

jim_thompson5910 · Answer

kinda like how there is a double root at -3 for x^2 + 6x +9

if you graph  x^2 + 6x +9, you'll see how it touches the x-axis and comes back

anonymous · Answer

yes I see, it just bounces of the x axis. is this always the case, meaning that if the graph just touches and doesn't cross the axis it is a double or more zero for the poly?

jim_thompson5910 · Answer

more like: if it touches the x-axis and comes back it is either a root of multiplicity 2, 4, 6, 8, ... ie a root of even multiplicity

anonymous · Answer

Great I got it. Thanks a million for your help! Big test tomorrow and struggled with this one question...

jim_thompson5910 · Answer

you're welcome, glad to be of help