Question

Determine the zeros of f(x) = x3 – 12x2 + 28x – 9

Answer 1

Factoring this function:
$$
f(x) = x^3 – 12x^2 + 28x – 9=(x-9) (x^2-3 x+1)
$$
Shows that the zeros occur when
$$
x-9=0\\
x^2-3 x+1=0
$$
x=9 is one zero. The other two can be found applying the quadratic formula to the second.

Answer 2

Make sense?

Answer 3

kind of. does it make sense? what does it mean by zeros? why do we look for zeros?

Answer 4

The zeros simply mean where does the function cross the x-axis. That is, when does the function equal zero. That's all.

Answer 5

When you have a function of power 3 as you do here, then the maximum number of REAL zeros (i.e. that aren't imaginary0, is also 3.

Answer 6

Sorry, I just came back. How exactly did you get (x-9)(x^2 +3x -1) ?

Answer 7

To find solutions to this cubic equation, start with the rational roots theorem. The possible roots are:
\[\pm \frac{ \text{factors of constant term} }{ \text{factors of leading coefficient} }= \pm \frac{ 1,3,9 }{ 1 }\]

Answer 8

Leading coefficient would be from the one with the highest degree?

Answer 9

I don't have a method that works every time, but I play around with factors that work. If I suspect a factor, I divide out that factor to get the other factors. Here, I suspected 9 was a factor.

Answer 10

Ah.

Answer 11

*(x-9) was a factor

Answer 12

yes. Try the following for x values: 1, -1, 3, -3, 9, -9 until one of them makes f(x) = 0. Then you would have found one root. Do synthetic division to find the quotient.

Answer 13

@ranga has stated it perfectly

Answer 14

@ybarrap, why can't it be (x^2 -9)? Therefore, I get -3 hisand +3 out of this

Answer 15

@ranga Got it.

Answer 16

But I just need to find the zeroes, not go deeper to find the quotent.

Answer 17

Try it, put in x=3 and x=-3 and see if f(x) = 0. If it does not, then they are not zeros.

Answer 18

Here the rational roots theorem gives you 6 possible roots: 1,-1,3,-3,9 and -9. One of them happens to work.
Thanks @ybarrap

Answer 19

That right, you will have three values of x that make f(x)=0 and there will be no more because you are limited by the power of 3 to a maximum of 3 zeros.

Answer 20

Got it. Thanks!

Answer 21

alright.

Answer 22

yw