Find a polynomial with integer coefficients that satisfies the given conditions.
R has degree 4 and zeros 3 − 4i and 5, with 5 a zero of multiplicity 2.
x = 3 - 4i
x = 3 + 4i
x = 5
x =5
two 5's because multiplicity 2
You always have to have the conjugate of an imaginary number.
(x -(3-4i))(x-(3+4i)(x-5)(x-5) = 0
Easiest way is to multiply the first two and the second two factors.
Then multiply those answers together to get the polynomial.
(x-5)(x-5)(x-3+4i)(x-3-4i) = 0
(x^2-10x+25)(x^2-6x+25)
multiply that out
complex values come in pairs (why?), so the roots are
3-4i, 3+4i, 5, 5
R(x) = (x-(3-4i))(x-(3+4i))(x-5)^2
= ((x-3)+4i)((x-3)-4i)(x-5)^2
= ((x-3)^2+4^2)(x-5)^2
= (x^2-6x+25)(x-5)^2
= x^4 - 16x^3 + 110x^2 - 400x + 625
Find a polynomial with integer coefficients that satisfies the given conditions.
R has degree 4 and zeros 1 − 3i and 2, with 2 a zero of multiplicity 2.
R(x) =
3-4i, 3+4i, 2, 2
Find a polynomial with integer coefficients that satisfies the given conditions.
R has degree 4 and zeros 2 − 4i and 5, with 5 a zero of multiplicity 2.
R(x) =
To find a polynomial that satisfies the given conditions, we can start by using the fact that if a complex number is a zero of a polynomial with integer coefficients, then its conjugate must also be a zero.
Given that 3 - 4i is a zero, we know that its conjugate, 3 + 4i, must also be a zero. We also know that 5 is a zero of multiplicity 2, meaning it is a repeated zero.
To find the polynomial, we can start by setting up the factors:
(x - (3 - 4i))(x - (3 + 4i))(x - 5)(x - 5) = 0
Simplifying this, we get:
((x - 3) + 4i)((x - 3) - 4i)(x - 5)^2 = 0
Expanding the first two factors, we get:
((x - 3)^2 - (4i)^2)(x - 5)^2 = 0
Simplifying further, we get:
((x - 3)^2 - 16i^2)(x - 5)^2 = 0
Since i^2 = -1, we can simplify the polynomial to:
((x - 3)^2 + 16)(x - 5)^2 = 0
Expanding the terms, we get:
(x^2 - 6x + 9 + 16)(x - 5)^2 = 0
Simplifying further, we get:
(x^2 - 6x + 25)(x - 5)^2 = 0
Multiplying everything out, we get:
(x^4 - 16x^3 + 101x^2 - 300x + 625) = 0
Therefore, the polynomial with integer coefficients that satisfies the given conditions is:
f(x) = x^4 - 16x^3 + 101x^2 - 300x + 625