Find a possible formula for a polynomial f with the following properties.
f has degree less-than-or-equal-to 2, f(0) = f(3) = 0 and f(5) = 30
f(x) = ?
k(x+0)(x+0)(x-3)
I know the zeroes for 3 of them are 0, 0, 3, how would i find out the rest?
so we have x-intercepts of 0 and 3
thus
f(x) = kx(x-3)
but (5,30) also lies on f(x), so ...
30 = k(5)(2)
10k = 30
k = 3
f(x) = 3x(x-3) or f(x) = 3x^2 - 9x
To find a possible formula for the polynomial f with the given properties, we can start by considering the fact that f(0) = 0 and f(3) = 0. This means that the factors (x - 0) and (x - 3) are roots of the polynomial.
Now, since the degree of f is less-than-or-equal-to 2, we know that the highest power of x in the polynomial will be x^2. So we can consider the general form of the polynomial as f(x) = k(x - 0)(x - 0)(x - 3).
However, we still need to determine the value of k, as well as the coefficient in front of x^2 to match the condition f(5) = 30.
To find the value of k, we can use the fact that f(5) = 30. Substituting x = 5 into the general form of the polynomial, we get f(5) = k(5 - 0)(5 - 0)(5 - 3) = k(5)(5)(2) = 50k. Since f(5) = 30, we can set up the equation 50k = 30 and solve for k. Dividing both sides by 50, we find that k = 30/50 = 3/5.
So the formula for the polynomial f(x) with the given properties is f(x) = (3/5)(x - 0)(x - 0)(x - 3). Simplifying this, we get f(x) = (3/5)x^2(x - 3), or f(x) = (3/5)x^3 - (9/5)x^2.
Therefore, a possible formula for the polynomial f with the given properties is f(x) = (3/5)x^3 - (9/5)x^2.