Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 5; zeros:1;-i; -7+1
To form a polynomial with the given degree and zeros, we'll first use the complex zeros and their conjugates to find the quadratic factors. Then we'll use the remaining zero to find the linear factor.
Given zeros:
1
-i
-7+1 (which simplifies to -6+1 = -5)
To find the quadratic factors, we'll use the complex zeros and their conjugates -i and +i:
(x - (-i)) = x + i
(x - i) = x - i
(x - (-5)) = x + 5
Now we can form the quadratic factors:
(x + i) * (x - i) = x^2 - i^2
(x + 5)
Multiplying the quadratic factors gives us the resulting quadratic polynomial:
(x^2 + 1) * (x + 5)
Now we'll find the linear factor using the remaining zero, 1:
(x - 1)
Now we can form the polynomial by multiplying all the factors together:
f(x) = (x^2 + 1) * (x + 5) * (x - 1)
Simplifying:
f(x) = (x^2 + 1)(x + 5)(x - 1)
So, the polynomial with degree 5 and the given zeros is f(x) = (x^2 + 1)(x + 5)(x - 1).