Question

Approximate the real number solution(s) to the polynomial function f(x) = x3 + 4x2 + x – 6

I got:
x = –3, x = –1, x = 2

Am I right?

Answer 1

\[x^3 + 4x^2 + x - 6 \implies (x - 1)(x + 2)(x + 3) \]

Answer 2

According to that information, we'd use the factor theorem.
In a factor \(x - k\), \(k\) is the root.

Answer 3

There are three factors, right? Let's make them all in the form of \(x - k\).\[x - 1 \\ x - (-2) \\ x-(-3) \]

Answer 4

Therefore, there are three roots namely:\[1\\-2\\-3 \]

Answer 5

@brittneyy18rm Can you use this method in all the other questions you have?

Answer 6

for the section I'm working on, yes. thank you!

Answer 7

You're welcome! :)

Answer 8

If the question is asking you to "Approximate" then I doubt they want you to factorise it. My guess would be that they want you to plot the curve and approximate where it crosses the x-axis.

Answer 9

That's a nice way too!

Answer 10

You could use this free online graphing calculator if want: https://www.desmos.com/calculator

It is fairly easy to use.

@ParthKohli rather than "nice way" I think this is the "expected way" to solve this. Factorising cubic polynomials is not a trivial operation and will not give "approximate" answers - it gives "exact" answers.

Answer 11

I am sorry. I am not that good at teaching. My apologies, sir.

Answer 12

no need to apologise ParthKohli - I just wanted to ensure you understood the difference.

and by the way, you are an EXCELLENT teacher in my view. :)

Answer 13

Thank you!

Answer 14

In my point of view ! Natural roots don't need to be approximated !

Answer 15

what do you mean Neemo?

Answer 16

Yes, I don't think that there are approximations for numbers like 1,-2,-3 :P

Answer 17

If the question was: "Find the real number solution(s) to the polynomial function...", then I would expect exact answers to be given.

However, the question here is stated as "Approximate the real number solution(s) to the polynomial function..." which leads me to believes that an approximation is required and hence some sort of graphical method.

And Parth, the curve will intersect the x-axis at 1, -2, -3, but, when drawing the graph you may not get those values exactly, which is why I believe it asks for approximations.

Answer 18

@brittneyy18rm - it would be nice to get your opinion on this as well. Have you been taught to factorise 3rd degree polynomials?

Answer 19

We don't need an approximation for Natural Roots ! Maybe the approximation is useful when we don't know the exact value of the real roots !
for example : x^3+x+1=0 !
and in this case ! If the drawing is correct ! He will certanily get the exact values ! :) !
The question is really confusing !

Answer 20

Neemo - are you saying that everyone ALWAYS plots ALL graphs PERFECTLY?

Answer 21

No ! I didn't say that ! ( I just said that these values are easy to see ! Maybe the drawing ! will just give us an idea about the roots !)
and Yes ! If they use a calculator ! or a "software" !