Pre Calculus
posted by Josh .
1. Find all rational zeros of the polynomial. Then determine any irrational zeros, and factor the polynomial completely. 3x^411x^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(x1)(3x^2  8x + 3)
The quadratic formula reveals the other two roots:
4/3 +/ √7/3
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If 2+i is a root, so is 2i, so x^2 4x + 5 is a factor.
So, f(x) = (x1)(x1)(x^2  4x + 5)