Question

Find all real zeros of the function.

f(x)=x^3-4x^2-11x+30

Answer 1

That cubic factors into:

(x+3)(x-2)(x-5)

setting all factors to zero, you get the zeros are: x = -3, 2, and 5.

Answer 2

@easyaspi314, you didn't show any factoring steps at all which does not help @Ammarah understand how to solve problems like this on their own.

Answer 3

@Hero question was for the zeros. Perhaps the student factored and wanted to check with someone else; or is allowed to use technology to find the zeros. Many possibilities.

Answer 4

You cannot make that assumption. The right course of action would have been to ask the student if he or she has factored it already.

Answer 5

Hero how do i factor it?

Answer 6

^Thus confirmation that the user has not factored it yet

Answer 7

@Hero If I show the factoring, student is more than welcome to ask to show how we get the factors. But I hear what you're saying.

Answer 8

im sorry please dont fight

Answer 9

wait how do i factor a cube root out?

Answer 10

To factor it, what you have to do is find one of the zeroes using the Rational Root Theorem. Once you do that, then you can use polynomial division or synthetic division to reduce the expression to a quadratic expression. Then from there you can finish factoring it.

Answer 11

@Ammarah, are you familiar with the rational root theorem?

Answer 12

so the zeros i found are 1,3,5,6,10,30

Answer 13

Those would be the possible zeroes. Also you should list them as
\[±1, ±2, ±3, ±5, ±6, ±10, ±15, ± 30\]

Answer 14

yeah but which ones do i use for synthetic division???

Answer 15

So all we need to do is find just one of these zeroes. I recommend using a calculator to program the function in to the calc.

That way you can easily compute

f(-1)
f(1)
f(-2)
f(2)

and so on until you find one of the zeroes.

Answer 16

You could do it by hand, but that would take up a lot of time.

Answer 17

You want to be able to find them quickly. I programmed the function in to my calculator and found that f(2) = 0 which means x - 2 is a factor

Answer 18

So then from there you could divide

\[\frac{x^3 - 4x^2 - 11x + 30}{x - 2}\] To get the quadratic expression.

Answer 19

And then finish factoring

Answer 20

@Hero

A student can just plot the cuboc and see the 3 zeros, and hence, see the three factors. And that's assuming the student is permitted to use a calulator for these types of questions (which they should be).

Answer 21

I understand @Easyaspi314. I'm fully aware of that. But, the student did ask how to factor it

Answer 22

got it

Answer 23

Got what @Ammarah

Answer 24

how to factor it u just list all the solutions....

Answer 25

*possible solutions

Answer 26

yea

Answer 27

In reality, we know that there are only three zeroes for the solution.

Answer 28

another question: Find all the zeros of the polynomial function.