Question

Questions about this equation f(x) = x^3 - x^2 - 4x + 4?

* How many zeros does this function have over the set of complex numbers?
* What is the maximum number of local extrema (maxima or minima) the graph of the function can have?
* What value does the function (y) approach as x approaches ∞ ?
* What value does the function (y) approach as x approaches - ∞ ?
* List the possible rational zeros of this function.
* Factor this polynomial completely over the set of complex numbers.

Answer 1

HERE IS WHAT I KNOW:
1) It is clear that there are 3 zeroes.
2) The max number of local exterma is 2, because this is a cubic polynomial.
3) The largest power we have here is 3, and term associated with it is x^3. Thus, y approaches infinity as x approaches infinity.
4) Similar to 3), y will approach negative infinity as x approaches negative infinity.

5 & 6 - ?

Answer 2

Not sure if this answers 5 and 6, but I know the zeros are:
x = 2
x = 1
x = -2

Answer 3

in this question, did you think about factoring by grouping?

Answer 4

I mean if you take x^2 as a common factor from the first 2 terms , and 4 as a common factor from the last 2 terms you will make it more simple

Answer 5

-4 Im sorry

Answer 6

So would it look like this? \[\frac{(x^3 - x^2 - 4x + 4) }{ (x^2 - 4) }\]\[(x-1) \]

Answer 7

And its fine :)

Answer 8

hmmmmmm that's not 100% right, I really think you got the idea
\[x ^{2} (x-1) -4 (x-1)=0\]
\[(x-1) (x ^{2}-4)=0\]

Answer 9

So would this be for
* List the possible rational zeros of this function.
OR
* Factor this polynomial completely over the set of complex numbers.

Answer 10

And in the end, \[(x - 1)(x^2 - 4) = 0\]\[(x-1)(x-2)(x+2) = 0\]

Answer 11

no, we just factored , there are another possible rational zeros I you were asked to find
but I do not know how to tell you how we can find them in English lol

Answer 12

Oh!

Answer 13

it's something about the constant term and the leading term

Answer 14

So is this better?
Factors of 1 (leading coefficient) are -1 and 1
Factors of 4 (coefficient of constant) are -1, 1, -2, 2, -4, 4
By rational root theorem, possible rational zeroes are -1, 1, -2, 2, -4, 4

Answer 15

I am sorry :(

Answer 16

Its alright! But is this the answer for
* List the possible rational zeros of this function.
OR
* Factor this polynomial completely over the set of complex numbers.

Answer 17

aha you got my point :)

Answer 18

the something you did is called finding rational possible zeroes, and the factor is what we did together above

Answer 19

Thank you!

Answer 20

okay you are welcome, but can I count on you to do the rest :P ?

Answer 21

Yes!

Answer 22

ok, good luck then :)

Answer 23

For 5) you must use the Rational Roots theorem which states that given for a given polynomial in the form:\[\bf P(x)=\sum_{k=0}^{n}a_kx^k=a_nx^n+a_{n-1}x^{n-1}+...+a_0x^0\]and if the polynomial has integral coefficients then its possible rational roots are given by:\[\bf Possible \ roots \ of \ P(x) = \frac{factors \ of \ a_0}{factors \ of \ a_n}\]This basically means that you are finding all the factors of the last constant in the polynomial and dividing them with all the factors of the leading coefficient. However many rational numbers you get could all be POSSIBLE roots.

Now for 6) they are asking you to factor the polynomial such that you include any complex factors, i.e. factors that are not purely Real. To factor the polynomial, we use factoring by grouping. The terms are already grouped for us so we just factor:\[\bf f(x)=x^3-x^2-4x+4 = x^2(x-1)-4(x-1)=(x^2-4)(x-1)\]\[\bf =(x-2)(x+2)(x-1)\]

Answer 24

@ilfy214

Answer 25

Oh wow. Thank you so much! This is perfect