what is a quartic function with only the two real zeros given
x=5 and x=1
y=-x^4-6x^3+6x^2-6x+5
y=x^4+6x^3-6x^2-5
y=x^4-6x^3+5x^2-6x+6
y=x^4-6x^3+6x^2-6x+5
can someone explain
To find a quartic function with the given real zeros, we can start by using the zero-product property.
The zero-product property states that if a product of factors is equal to zero, then at least one of the factors must be equal to zero.
In this case, the two real zeros are x = 5 and x = 1. Therefore, we can start by setting up two linear factors in the form (x - r), where r represents the root (zero).
So, for the given zeros, we have:
(x - 5) = 0
(x - 1) = 0
Now, we can multiply these two factors together to get a quadratic function:
(x - 5) * (x - 1) = 0
Expanding this expression, we have:
x^2 - 6x + 5 = 0
However, we are looking for a quartic function, so we need to find two more roots (zeros). To obtain these, we can either factor the quadratic equation or use the quadratic formula.
In this case, the quadratic equation can be factored by considering the sum and product of the roots.
The sum of the roots is -(-6) = 6, and the product of the roots is 5.
Since there are two more roots, let's assume they are p and q.
The sum of all four roots is equal to the coefficient of the x^3 term divided by the coefficient of the x^4 term, which is 6/1 = 6.
Therefore, p + q + 5 + 1 = 6.
Simplifying, p + q = 0.
The product of all four roots is equal to the constant term divided by the coefficient of the x^4 term, which is 5/1 = 5.
Therefore, p * q * 5 * 1 = 5.
Simplifying, p * q = 1.
Given p + q = 0 and p * q = 1, we can solve these simultaneous equations to find p and q values.
By solving p + q = 0, we have p = -q.
Substituting this into p * q = 1, we get q * (-q) = 1, which simplifies to -q^2 = 1.
Solving for q, we have q = √(-1).
Since we need real zeros, we can see that q is an imaginary number. Therefore, the original claim of having only two real zeros is incorrect. A quartic function should have four roots, which may or may not be real.
So, the correct answer regarding a quartic function with only the two given real zeros, x = 5 and x = 1, is that it is not possible to find such a function.