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
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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)

To find the rational zeros of a polynomial, we can use the Rational Root Theorem.

1. Rational Zeros of the Polynomial:

The Rational Root Theorem states that any rational zero of a polynomial with integer coefficients will have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In this case, the constant term is 3 and the leading coefficient is also 3. The potential rational zeros will have the form p/q, where p is a factor of 3 and q is a factor of 3. The factors of 3 are ±1 and ±3.

So the possible rational zeros are: ±1, ±3.

We can now use synthetic division or polynomial long division to check if any of these potential zeros are actual zeros. By checking these values, we find that none of the possible rational zeros are actually zeros of the polynomial.

2. Irrational Zeros and Factoring:

To find the irrational zeros and factor the polynomial completely, we need to use other methods. One method is to use the quadratic formula to find the zeros of the quadratic expression left after factoring out any rational zeros.

In this case, since the degree of the polynomial is 4, we can factor out any remaining rational zeros by using synthetic division. After dividing by the rational zeros, we are left with a quadratic polynomial:

(3x^2 + 2x - 3).

To find the zeros of this quadratic polynomial, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a.

For this quadratic polynomial, a = 3, b = 2, and c = -3. Substituting the values into the formula:

x = (-2 ± √(2^2 - 4(3)(-3))) / (2(3)).

x = (-2 ± √(4 + 36)) / 6.

x = (-2 ± √40) / 6.

Simplifying further:

x = (-2 ± 2√10) / 6.

x = (-1 ± √10) / 3.

So the irrational zeros of the polynomial are (-1 + √10) / 3 and (-1 - √10) / 3.

To factor the polynomial completely, we can now write the factored form as:

3x^4 - 11x^3 + 5x^2 + 3x = (x - (-1 + √10) / 3)(x - (-1 - √10) / 3) multiplied by any irreducible factors if present.