find the complex zeros of the polynomial. Write f in factored form: F(x)= 2x^2-5x^3-20x^2+115x-52=0
2x^2-5x^3-20x^2+115x-52=0
-5x^3 - 18x^2 + 115x - 52 = 0
5x^3 + 18x^2 - 115x + 52 = 0
no obvious NICE factors
the graph suggests 3 real roots, thus no complex zeros
http://www.wolframalpha.com/input/?i=5x%5E3+%2B+18x%5E2+-+115x+%2B+52+%3D+0
I suspect a typo
To find the complex zeros of the polynomial and write it in factored form, we need to follow these steps:
Step 1: Rearrange the equation to set it equal to zero:
2x^2 - 5x^3 - 20x^2 + 115x - 52 = 0
Step 2: Group the terms:
(x^2 - 5x^3) + (-20x^2 + 115x) - 52 = 0
Step 3: Factor out the greatest common factor (GCF) from each group of terms:
x^2(x - 5) - 23x(5 - x) - 52 = 0
Step 4: Simplify each group of terms:
x^3 - 5x^2 - 115x + 115x - 52 = 0
Step 5: Combine like terms:
x^3 - 5x^2 - 52 = 0
Step 6: Now, to find the complex zeros, we can use different methods such as synthetic division or factoring. Let's try factoring:
Step 7: Set the equation equal to zero:
x^3 - 5x^2 - 52 = 0
Step 8: Try different factors of the constant term (-52) to find the roots. In this case, we find that x = 2 is a factor.
Step 9: Perform long division to obtain a quadratic equation:
(x - 2)(x^2 - 3x + 26) = 0
Step 10: Use the quadratic formula or factor the quadratic equation (x^2 - 3x + 26) to find the remaining complex zeros.
The complex zeros of the polynomial are:
x = 2 (real zero)
x = (3 ± √(-59))/2 (complex zeros)
Therefore, the factored form of the given polynomial is:
F(x) = 2(x - 2)(x - [(3 ± √(-59))/2])