Question

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

Answer 1

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
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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)