R has degree 4 and zeros 5 − 3i and 4, with 4 a zero of multiplicity 2.?
Find a polynomial with integer coefficients that satisfies the given conditions.
R has degree 4 and zeros 5 − 3i and 4, with 4 a zero of multiplicity 2.
AND
Find a polynomial with integer coefficients that satisfies the given conditions.
T has degree 4, zeros i and 1 + i, and constant term 12.
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If 5 − 3i is a zero, then 5 + 3i must also be a zero.
Use the factor theorem to convert these four zeros into factors:
(x − 5 + 3i) (x − 5 − 3i) (x − 4) (x − 4)
Multiply these factors together:
x^4 − 18x^3 + 130x^2 − 432x + 544
If i is a zero then -i is also a zero. If 1 + i is a zero, then 1 - i is also a zero.
Use the factor theorem to convert these four zeros into factors:
(x − i) (x + i) (x − 1 − i) (x − 1 + i)
Multiply these factors together:
x^4 − 2x^3 + 3x^2 − 2x + 2
Multiply by 6 to achieve the desired constant term:
6x^4 − 12x^3 + 18x^2 − 12x + 12