The graph of a rational function has a local maxima at (-1,0) and (8,0). The complex number 2+3i Is a zero of the function. What is the least possible degree of the function
Answer is 6
But i need help with an explanation.
Since the maxima are on the x-axis, they must be roots of even multiplicity. There must be a conjugate root as well, so that makes at least 6 roots.
To find the least possible degree of the function, we need to consider the number of zeros the function can have.
Given that the graph of the rational function has a local maximum at (-1,0) and (8,0), it means that the function must have a factor of (x + 1)^2 and (x - 8)^2. This is because the function needs to touch the x-axis at (-1,0) and (8,0), which results in the local maxima.
Additionally, we are given that the complex number 2 + 3i is a zero of the function. Since complex zeros occur in conjugate pairs, we know that the function must also have a factor of (x - 2 - 3i) and (x - 2 + 3i).
Now, to find the least possible degree, we multiply all these factors together:
(x + 1)^2 * (x - 8)^2 * (x - 2 - 3i) * (x - 2 + 3i)
Expanding this equation will give us a polynomial function of degree 6. Therefore, the least possible degree of the function is 6.