Final answer:
To write a polynomial equation P with roots 5, -7, 3+i, and 3-i, we create factors for each root and multiply them together. For the complex roots, we multiply the conjugate pair to get a quadratic with real coefficients, resulting in the polynomial P(x) = (x - 5)(x + 7)(x^2 - 6x + 10).
Step-by-step explanation:
To write an equation P in the lowest degree with the given roots 5, -7, 3+i, and 3-i, we need to form factors from these roots and then multiply them together. For real roots, the factors are (x - root). For the complex roots, because they come in conjugate pairs, their factors multiply to give a quadratic with real coefficients. The factor for 3+i and 3-i will be (x - (3+i))(x - (3-i)) which simplifies to (x - 3 - i)(x - 3 + i), i.e. (x - 3)^2 + 1.
The complete polynomial P(x) is obtained by multiplying all the factors:
P(x) = (x - 5)(x + 7)((x - 3)^2 + 1).
To find the polynomial in the lowest degree, we expand and simplify the product of these factors:
P(x) = (x - 5)(x + 7)(x^2 - 6x + 10).