Form a polynomial f(x) with 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 degree and zeros, we can use the fact that if a number is a zero of a polynomial, then the polynomial has a factor of (x - zero).
Given the zeros: -3, -i, -6+i, we can write the factors as:
(x - (-3)) = (x + 3)
(x - (-i)) = (x + i)
(x - (-6 + i)) = (x + 6 - i)
Now, since the polynomial has a degree of 5, we need to multiply these factors together with two more factors.
Since we know that the coefficients of the polynomial are real, the complex conjugates of -i and -6+i will also be zeros. Therefore, the factors become:
(x - (-3)) = (x + 3)
(x - i)(x + i) = (x^2 + 1)
(x - (-6 + i))(x - (-6 - i)) = (x - 6 + i)(x - 6 - i) = (x - 6)^2 - (i)^2 = (x - 6)^2 + 1
Multiplying these factors together, we get:
F(x) = (x + 3)(x^2 + 1)(x - 6)^2 + 1
Therefore, the polynomial is:
F(x) = (x + 3)(x^2 + 1)(x - 6)^2 + 1