Write a polynomial function of least degree with rational coefficients so that P(x)=0 has the given root
3-7i
P(x)= x^2 - __x +__
complex roots come in conjugate pairs, so if one is
3-7i, the other is 3+7i
sum of roots = 3-7i + 3+7i = 6
product of roots = (3-7i)(3+7i)
= 9 - 49i^2 = 9 + 49 = 58
equation is
x^2 - 6x +58 = 0
To find a polynomial function with rational coefficients that has a given root, we need to remember that complex roots always come in conjugate pairs. Since the given root is 3-7i, the conjugate root is 3+(-7i) = 3+7i.
To construct a polynomial with these roots, we can multiply two factors. One factor will be (x - 3 + 7i) since it corresponds to the first root, and the other factor will be (x - 3 - 7i) since it corresponds to the conjugate root.
Next, we multiply the factors to find the polynomial:
(x - 3 + 7i)(x - 3 - 7i)
Using the difference of squares formula [(a-b)(a+b) = a^2 - b^2 ]:
= (x - 3)^2 - (7i)^2
= (x - 3)^2 - 49i^2
Since i^2 is equal to -1:
= (x - 3)^2 + 49
Therefore, the polynomial function of least degree with rational coefficients that has the root 3-7i is:
P(x) = (x - 3 + 7i)(x - 3 - 7i)
Expanded form: P(x) = x^2 - 6x + 9 + 49
Simplified form: P(x) = x^2 - 6x + 58