Determine the zeros of f: [help!]?

Question

Let f(x) = -3(x+2)(x-4)(x-2)(x+6)


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so i need to find the zeros of that, i expanded it and put it in the form: 3(5x^2 + 16x -12) ... i dont know if that's right. now how do i find the zeros? the answer is -6, -2, 2 and 4.

how do i get it? show work plz!! thx for the help..


Also, how do you find the maximum number of turning points ??

btw , yes i realize that its the answers the same as in the brackets but dont u have to expand it first?

Answer 1

No, you don't have to expand. You could, of course, but that would make the problem somewhat more complex; you'd end up with a fourth order function. Simply find the numbers that make your function equal to zero.

x +2 = 0 --> x = -2
x - 4 = 0 --> x = 4
x - 2 = 0 --> x = 2
x + 6 = 0 --> x= -6

Answer 2

ZEROs are the values of X where f(x) will result in zero.

In this case, -3 will stay constant. The only thing that can change is X. If you take each pairs of parenthesis and solve it for x where it will be zero, you'll find all the zeros.

Think about this.
Y=-3(A)(B)(C)

if you make A, B, or C zero, Y will be zero, correct?
Then if A=x+2, then x=-2 will yield zero.

No, you got it backwards.... you have to FACTOR THEM to find zeros. In another words, if it comes in x^2+Bx+C, then you'll have to FACTOR THEM to put it in (X+a)(X+b) format.

As to turning points....

Notice you have (x+/- something) 4 times. If you multiply this out, you will end you with x raised to the 4th power. If you got x^2, you did something wrong.

Now, think about what ZEROs really are. It is the point where graph crosses X axis. In another words, the line will bob up and down and crosses 4 times. How many times does it have to change directions before it can do that?

Draw it out and find out for yourself.
Trust me, it's a lot easier to understand when you ROUGHLY sketch this out.

Can you come to a generalization between number of X(es) in the factored form, what happens when you multiply them out, and the number of turning points?

Tony below....
Zeros are not the same thing as "turning points" or "inflection points". What you are thinking of, is a zeros for the first derivative of the function. Then, the zeros of the f'(x) (f prime of x) would be the inflection points. That, actually is not the question being asked here.

Answer 3

You don't expand to get the zeros. The zeros of f(x) just mean when will f(x) equal zero. This will happen when any one of the term equals zero itself (e.g the [x-2] term), as then multiplying the other terms will lead to a final result of zero.

So, when will each of the terms equal zero? x+2 = 0 when x= -2, so continue on this track for each term and you get x=-2,4,2,-6.

The max number of turning points is always one less than the degree of the function. ie, you have four x's in your function so if you bothered to expand it the highest term would be x^4 (see, you don't need to expand it to figure out this part). So, max turning points is 4-1=3. Think of a basic parabola - it's y = x^2 and has just one turning point, so it's highest term is 2, and 2-1=1 so 1 turning point.

Answer 4

You are making it much harder than it is. Look: f(x) = 0 if and only if -3(x + 2)(x - 4)(x - 2)(x + 6) = 0. But a product of factors is zero if and only if one of the factors = 0. -3 cannot be 0. x + 2 = 0 if x = -2, x - 4 = 0 if x = 4, and so on. See! No expansion needed.

What is a "turning point"? If turning point is a fuzzy name for zeros, the max for a polynomial is the degree of the polynomial; if turning points are max or min, the max number is one less than the degree; if inflection points (points where the concavity changes), the max is two less than the degree.

Answer 5

No, you don't have to expand it first. Just plug in the value -6 for x. Then:
f(x) = -3(something)(something)(something)(-6 + 6)
The last term is zero. It doesn't matter what the rest of the expression is, because any number multiplied by zero, is zero.

And the same for x = -2, x= 2, and x= 4.

Answer 6

mystery_girl07--this reply is not directed toward you or your question, but Tony above who answered it and was extremely rude on my question. I'm sorry if there's confusion.

Tony, I'm only answering this question to inform you of how rude you were, and because you have no questions. ******* scum like you, I have NO idea how you're a "top constructor" of mathematics, but it seems like to me you're a "top constructor" of rudeness, obscenity, and a foul mouth. How. Dare. You. I was merely going to award you points and a best answer if you answered all the questions, but with a sick mind like yours I suppose you're probably a preditor who's on here to solicit innocent people. I've already taken the liberty of reporting you, so don't worry. In the way you answered this persons question I believe you are more than capable at answering math, but not so capable of being nice and polite. It's scum like you who make yahoo! answers worse. The way you answered my question was uncalled for. Go jump off a cliff--and save the world from your rudeness.

Answer 7

u poor thing