Form a polynomial, f(x), with real coefficients having the given degree and zeros.
Degree 3; zeros: 1 + i and -10
Hint: If x = 1+i is a root, x = 1 -i must also be a root. Comples roots occur in pairs.
So (x+10), (x- 1 -i) and x -1 + i) must be factors.
Multiply all that out for a third order polynomial.
To form a polynomial with the given degree and zeros, we can use the fact that the zeros of a polynomial are the values of x for which the polynomial evaluates to zero.
Since the polynomial has degree 3 and the given zeros are 1 + i and -10, we know that the factors of the polynomial are (x - (1 + i)), (x - (-10)).
To find the remaining factor, we need to consider the conjugate of 1 + i, which is 1 - i. So, the third factor of the polynomial is (x - (1 - i)).
Multiplying these three factors together will give us the polynomial.
First, let's simplify the factors:
(x - (1 + i)) = x - 1 - i
(x - (-10)) = x + 10
(x - (1 - i)) = x - 1 + i
Now, let's multiply the factors:
f(x) = (x - 1 - i)(x + 10)(x - 1 + i)
Expanding this expression will yield the polynomial:
f(x) = (x^2 - x - xi + 10x - 10 - 10i)(x - 1 + i)
= (x^2 + 9x - 10 - (1 + xi) - 10i)(x - 1 + i)
= (x^2 + (9 - xi)x - (10 + 1 + xi - 10i - i))(x - 1 + i)
= (x^2 + (9 - xi)x - (11 + xi - 10i))(x - 1 + i)
= x^3 + x^2(9 - xi) - x(11 + xi - 10i) - (11 + xi - 10i)(x - 1 + i)
Now, simplify the expression further to obtain the final polynomial.