Form a polynomial f(x) with the real coefficients having the given degree and zeros.
Degree 5; Zeros: -3; -i; -6+i
f(x)=a( )
To form a polynomial with the given zeros and degree, we need to know that complex roots always come in conjugate pairs. Since we are given the zero -i, its conjugate, i, is also a zero.
The degree of the polynomial is given as 5. Since complex zeros have a multiplicity of 1, and the real zero -3 has a multiplicity of 1, there are no repeated zeros. Therefore, we need to form a polynomial with the factors corresponding to each zero.
The factor corresponding to the zero -3 is (x + 3).
The factor corresponding to the zero -i is (x + i).
The factor corresponding to the zero i is (x - i).
The factor corresponding to the zero -6 + i is (x - (-6 + i)), which simplifies to (x + 6 - i).
Now, we can form the polynomial by multiplying all these factors:
f(x) = (x + 3)(x + i)(x - i)(x + 6 - i)
Expanding this expression, we get:
f(x) = (x + 3)(x^2 + 1)(x + 6 - i)
Multiplying further:
f(x) = (x^3 + 4x^2 + 3x + 9)(x + 6 - i)
Simplifying:
f(x) = x^4 + 10x^3 + (25 - i)x^2 + (45 - 6i) x + 54 - 9i
Therefore, the polynomial f(x) with the given zeros and degree is:
f(x) = x^4 + 10x^3 + (25 - i)x^2 + (45 - 6i) x + 54 - 9i