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?
I know The correct answer is 6. I need help with the explanation please.
Note that the two maxima lie on the x-axis. So, they are double roots. Both are real.
Now add to that the complex root and its conjugate, and you have degree 6.
To find the least possible degree of the rational function, we need to consider the given information:
1. The graph has a local maxima at (-1,0) and (8,0): This implies that the function has a horizontal asymptote at y = 0 and the function crosses the x-axis (has x-intercepts) at -1 and 8.
2. The complex number 2+3i is a zero of the function: This means that the function has a factor of (x - (2+3i)) in its equation.
Based on this information, let's construct the equation of the rational function step by step.
Since the graph has a local maxima at (-1,0) and (8,0), the function can be written as a product of linear factors as follows:
F(x) = a(x + 1)(x - 8)
Here, a is a constant that multiplies the function.
Next, as the given complex number 2+3i is a zero of the function, we can include this factor in the equation as well:
F(x) = a(x + 1)(x - 8)(x - (2+3i))
To ensure that F(x) is a rational function (the denominator is not zero), we also need to include the conjugate of the complex zero:
F(x) = a(x + 1)(x - 8)(x - (2+3i))(x - (2-3i))
Expanding this equation gives:
F(x) = a(x + 1)(x - 8)((x - 2) - 3i)((x - 2) + 3i)
F(x) = a(x + 1)(x - 8)((x - 2)^2 - (3i)^2)
F(x) = a(x + 1)(x - 8)(x^2 - 4x + 4 + 9)
Simplifying further:
F(x) = a(x + 1)(x - 8)(x^2 - 4x + 13)
To find the least possible degree of the function, we count the number of factors in the equation. Here we have four factors: (x + 1), (x - 8), (x^2 - 4x + 13), and a.
Therefore, the least possible degree of the function is 4.
However, the correct answer is given as 6, which means there are two additional factors that we haven't considered yet.
Since the graph has a local maxima at (-1,0) and (8,0), the function must touch but not cross the x-axis at these points. This can be achieved by squaring the factors (x + 1) and (x - 8) to make them positive:
F(x) = a(x + 1)^2(x - 8)^2(x^2 - 4x + 13)
Now, when we expand this equation, we get:
F(x) = a(x^2 + 2x + 1)(x^2 - 16x + 64)(x^2 - 4x + 13)
Simplifying further:
F(x) = a(x^2 + 2x + 1)(x^4 - 20x^3 + 169x^2 - 64x - 832)
Now, the equation contains six factors: (x + 1)^2, (x - 8)^2, (x^2 + 2x + 1), (x^2 - 16x + 64), (x^2 - 4x + 13), and a.
Therefore, the least possible degree of the function is 6.
Hence, the rational function has a least possible degree of 6.