Answer:
x⁴ -3x³ -x² -27x -90
Step-by-step explanation:
We generally study polynomials that have real (and rational) coefficients. One of the features of such polynomials is that irrational and/or complex roots come in conjugate pairs. That means the root -3i is always accompanied by the root +3i if the polynomial has rational coefficients.
For any polynomial, if q is a zero, then (x -q) is a factor.
Knowing that, you know the minimum polynomial with these roots will have factors ...
(x -(-2)), (x -5), (x -(-3i)), and (x -3i)
The last two of these are recognizable as factors of the difference of squares:
a² -b² = (a -b)(a +b)
(x -3i)(x +3i) = x² -(3i)² = x² -(-9) = x² +9
Now, you know the factored polynomial is ...
y = (x +2)(x -5)(x² +9)
You can use the distributive property repeatedly to expand this to standard form.
y = (x² -3x -10)(x² +9) = x²(x² -3x -10) +9(x² -3x -10)
y = x⁴ -3x³ -10x² +9x² -27x -90
y = x⁴ -3x³ -x² -27x -90
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Additional comment
For the purpose here, an irrational root will be of the form (for rational a, b, c) ...
a +b√c . . . . where c is not a perfect square
and a complex root will be of the form ...
a +bi
The conjugate of either of these forms is obtained by changing the sign:
a -b√c or a -bi
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You may notice that the form a±b√c will include the form a±bi when c is negative. So, the conjugate pairs we're talking about are this form with either of c < 0 or c not a perfect square, or both.
