Find a possible formula for the polynomial with these properties: f is the third degree with f(-3)=0, f(1)=0, f(4)=0, f(0)=4.
f(x)=?
*I don't even know where to start. Please help, step by step if possible.
f is the third degree with f(-3)=0, f(1)=0, f(4)=0, f(0)=4.
f is the third degree, in general case we have
f(x)=ax^3+bx^2+cx+d
f(-3)=0
f(-3)=a(-3)^3+b(-3)^2+c(-3)+d
f(-3)=-27a+9b-3c+d, then
-27a+9b-3c+d=0 (I)
f(1)=0
f(1)=a*1^3+b*1^2+c*1+d
f(1)=a+b+c+d, then
a+b+c+d=0 (II)
f(4)=0
f(4)=a*4^3+b*4^2+c*4+d
f(4)=64a+16b+4c+d, then
64a+16b+4c+d=0 (III)
f(0)=4
f(0)=a*0^3+b*0^2+c*0+d
f(0)=d, then
d=4 (IV)
Form (I), (II), (III), (IV) we need to solve the system
-27a+9b-3c+d=0
a+b+c+d=0
64a+16b+4c+d=0
d=4
and find a, b, c
The function f, which is actually called f(x) has 3 roots
so when it says f(-3) = 0
that means at x = -3, there is no y value, so that is one point
the same goes for 1 and 4, those are also "zeroes" which are places where the function, f, touches the x-axis
and as for f(0) = 4, another point is at (0,4)
Now, from these zeroes, -3, 1, and 4, we can get the equation which is
(x+3)(x-1)(x-4)
And so when you multiply it out you get an x^3, which means it is a third degree polynomial, always remember that the highest exponent tells you how many roots/"zeroes" there are and what degree the equation is
I can't explain why it is, but if anyone here can, please feel free to do so...
Math_helper's method is entirely correct, but realintegerz is very very close to a much easier way to do this.
An alternative form of the cubic is A(x+B)(x+C)(X+D). It isn't quite as general as the one math_helper posted but is perfectly sufficient for this question.
$\displaystyle f(-3) = 0 \implies A(-3+B)(-3+C)(-3+D)=0 \implies$ one of the factors is 0, using the null factor theorem. Without loss of generality we can say that this factor is -3-B = 0, giving B = 3.
Repeating this for the remaining zeroes gives C=-1 and D = -4. We then solve for A in the usual manner.
When you are using this to solve a problem simply write down the factors as realintegerz did, but don't forget the constant A
f(x) = A(x--3)(x-1)(x-4)
=A(x+3)(x-1)(x-4)
Then use the
remaining peice of information f(0) = 4 to find A
f(0) =4
A3(-1)(-4) = 4
A = 1/3
So the solution, which is equivalent to that found by math_helper is
f(x) = (1/3)(x+3)(x-1)(x-4)