Final answer:
To find the polynomial with the given roots -1 and 5-2i, we must include the conjugate root 5+2i and multiply the factors (x + 1), (x - 5 + 2i), and (x - 5 - 2i) to get the polynomial x^3 - 9x^2 + 19x + 29.
Step-by-step explanation:
To find a polynomial with real coefficients that has the roots -1 and 5-2i, we must also include the conjugate of 5-2i, which is 5+2i, because complex roots of polynomials with real coefficients always come in conjugate pairs. The factors corresponding to these roots would be (x + 1), (x - (5-2i)), and (x - (5+2i)).
Multiplying these factors together gives us the desired polynomial:
- (x + 1)
- (x - 5 + 2i)
- (x - 5 - 2i)
First, combine the two factors with complex numbers:
(x - 5 + 2i)(x - 5 - 2i) = x^2 - 5x - 2ix - 5x + 25 + 10i + 2ix - 10i - 4i^2 = x^2 - 10x + 25 + 4 = x^2 - 10x + 29
Now, multiply this result with the remaining linear factor:
(x + 1)(x^2 - 10x + 29) = x^3 - 10x^2 + 29x + x^2 - 10x + 29
Combining like terms, we obtain:
x^3 - 9x^2 + 19x + 29
This is the polynomial with real coefficients that has the given roots.