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?
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. >.>
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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