1. Supposed that a polynomial function of degree 6 with rational coefficients has .√5, 6+4i and 3+√2 as zeros. Find the other zeros .
2. Find a polynomial function of the lowest degree with rational coefficients and the following as some of its zeros: -√8,3i.
Please Help with both questions and list out your answer please !
2. Find a polynomial function of the lowest degree with rational coefficients and the following as some of its zeros: -√8,3i.
Please Help with both questions and list out your answer please !
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1. The other zeros are the conjugates of the 3
numbers you wrote.
It's a theorem that if a+bi is a root of a polynomial
with real coefficients, so is its complex conjugate.
So one of the other roots is 6-4i.
Also if the polynomial has rational coefficients
and a+b√d (d not a square) is a root, so is
a-b√d. So the other 2 roots are -√5 and 3-2√2
2). Same principle! Two of the other zeros are √8 and -3i
So the polynomial of lowest degree with rational coefficients that has
all these as the roots has degree 4 and is
(x-√8)(x+√8)(x-3i)(x+3i) = (x^2-8)(x^2+9) = x^4 + x^2 -72.
numbers you wrote.
It's a theorem that if a+bi is a root of a polynomial
with real coefficients, so is its complex conjugate.
So one of the other roots is 6-4i.
Also if the polynomial has rational coefficients
and a+b√d (d not a square) is a root, so is
a-b√d. So the other 2 roots are -√5 and 3-2√2
2). Same principle! Two of the other zeros are √8 and -3i
So the polynomial of lowest degree with rational coefficients that has
all these as the roots has degree 4 and is
(x-√8)(x+√8)(x-3i)(x+3i) = (x^2-8)(x^2+9) = x^4 + x^2 -72.