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# Polynomials

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The polynomial  has degree 3. If f(-1)=15, f(0)=0, f(1)=5 and f(2)=12, what are the x intercepts of the graph of f?

Nov 28, 2018

#1
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So let's dissect this problem, with the info we are given.

The degree of the polynomial is three, which means that there is at most ax^3, a being a coefficient, typically just one.

The problem tells you a set of coordinates, in a different manner. Instead of (x,y)... they give it to you in a statement.

The simplest way to think about this is the template f(x) = y.

OR

Stating if the function plugs in x then it equals y,

OR (how you will hear it most commonly in class)

"f of x" (fancy way of saying f(x) but comes in handy) equals y coordinate.

"If f(-1) = 15" means the ordered pair (-1,15)

"f(1) = 5" means (1,5)

"f(2) = 12" means (2,12)

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The question is asking to find x intercepts which means you'll be factoring your polynomia when you find out what it even... is.

To do this, think of how a graph x^3 is graphed, just for visually plotting all your points.

I'm sure someone else can continue this because, I am short on time.

Hope this helps in some regard c:

P.M. me if you need more assistance, I'm always on my computer and can always use a math break. Except that's oxymoron to what I just mentioned. You get the point. >.>

Nov 28, 2018
#2
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We have the form

y = ax^3 + bx^2 + cx + d

Because f(0) = 0....then d = 0...and we can solve this system

a(-1)^3 + b(-1)^2 + c(-1) = 15

a(1)^3 + b(1)^2 + c(1) = 5

a(2)^3 + b(2)^2 + c(2) = 12          which translates to

-a + b  - c  =  15

a  + b  + c  =   5

8a + 4b + 2c =  12

Adding the first  two equations produces  2b = 20 ⇒  b = 10

Using the last two  equations, we have

a + 10 + c = 5    ⇒  a + c =  - 5

8a + 40 + 2c = 12 ⇒  8a + 2c = -28 ⇒  4a + c = -14

Multiply the last equation by -1 and add to the frist equation

-4a - c = 14

a + c = 10

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-3a = 24   ⇒ a = -3

And a + c = -5 ⇒  c = -2

So....the polynomial is

-3x^3 + 10x^2 - 2x    factor this and set to 0

x ( -3x^2 + 10x - 2) = 0

We  know that one root is  x = 0   [ this is one x intercept ]

So we need to solve this

-3x^2 + 10x - 2  = 0       multiply through by  -1

3x^2  - 10x + 2 = 0

Using the quadratic formula the other two x intercepts are

5 ±√19

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3   Nov 28, 2018