4.
a) Determine the general equation for the family of degree 6 polynomial functions having zeros at x = 3 (order 2) , x = -1 (order 1) and x = 4 (order 3). (1 mark)
b) Write the equation of the above function, in factored form, passing through the point (2, 48) (2 marks)
a) To determine the general equation for the family of degree 6 polynomial functions with the given zeros and their orders, we need to construct a polynomial equation that satisfies each zero and its corresponding order.
Given the zeros at x = 3 (order 2), x = -1 (order 1), and x = 4 (order 3), we can express the equation as follows:
(x - 3)^2 * (x + 1) * (x - 4)^3 = 0
This equation represents a polynomial with a degree of 6, which has the zeros and their respective orders specified.
b) To find the equation of the function in factored form passing through the point (2, 48), we can use the general equation obtained in part a and substitute the coordinates of the given point into the equation.
Let's substitute x = 2 and y = 48 into the general equation:
(2 - 3)^2 * (2 + 1) * (2 - 4)^3 = 0
Simplifying this equation, we get:
(-1)^2 * (3) * (-2)^3 = 0
1 * 3 * (-8) = 0
-24 = 0
Since -24 does not equal 0, the equation is not satisfied by the point (2, 48).
Therefore, there seems to be an error or inconsistency in the given information, as it is not possible for the point (2, 48) to lie on the curve defined by the general equation with the given zeros and their orders.
a) To determine the general equation for the family of degree 6 polynomial functions with the given zeros and their orders, we start by considering each zero separately.
The zero x = 3 has an order of 2, so it appears twice in the factors of the polynomial equation. Therefore, one of the factors is (x - 3)^2.
The zero x = -1 has an order of 1, so it appears once in the factors of the polynomial equation. Therefore, another factor is (x + 1).
The zero x = 4 has an order of 3, so it appears three times in the factors of the polynomial equation. Therefore, another factor is (x - 4)^3.
Now we can multiply these factors together to get the general equation:
(x - 3)^2 * (x + 1) * (x - 4)^3
b) To find the specific equation passing through the point (2, 48), we need to determine the value of the constant term in the equation. We substitute the coordinates of the point into the equation and solve for the constant:
(2 - 3)^2 * (2 + 1) * (2 - 4)^3 = 48
(-1)^2 * 3 * (-2)^3 = 48
1 * 3 * -8 = 48
-24 = 48
This is not a true statement, indicating that there is an error. Please recheck the given information or calculations.