posted by Josh .
1. Find all rational zeros of the polynomial. Then determine any irrational zeros, and factor the polynomial completely. 3x^4-11x^3+5x^2+3x
2. Find the polynomial with leading coefficient 1 that has a degree of 4, a zero of multiplicity 2 at x=1 and a zero at x=2+i
We see right off that x=0 is a root.
Any rational roots must have a numerator which divides 3, and a denominator which divides 3. So, try 3,1,1/3
Then we have x(x-1)(3x^2 - 8x + 3)
The quadratic formula reveals the other two roots:
4/3 +/- √7/3
If 2+i is a root, so is 2-i, so x^2 -4x + 5 is a factor.
So, f(x) = (x-1)(x-1)(x^2 - 4x + 5)