which function below, if either, has zeros that are complex conjugates?
P(x)= X^4-7x^3-27x^2+63x+81
H(x)=x^3-x^2+25x-25
H(x) = x^2(x-1) + 25(x-1)
= (x-1)(x^2 + 25)
for zeros,
(x-1)(x^2 + 25) = 0
x = 1, or x = ±5i
±5i are complex conjugates
Off hand, I can't see any for the first one.
what do you mean
complex conjugates are numbers of the form
a+bi and a-bi
±5i can be written as
0+5i and 0-5i, which fits the above form.
To determine if a function has zeros that are complex conjugates, we need to check if the coefficients of the function are real and if the non-real zeros occur in conjugate pairs.
Let's examine the coefficients of each function:
For the function P(x) = x^4 - 7x^3 - 27x^2 + 63x + 81:
- The coefficients of all the terms (-7, -27, 63, and 81) are real numbers.
Now, let's consider H(x) = x^3 - x^2 + 25x - 25:
- Again, all the coefficients (-1, 1, 25, and -25) are real numbers.
To determine if either function has complex conjugate zeros, we need to find the zeros of each function. However, directly finding the zeros of quartic and cubic polynomials can be challenging and time-consuming.
A more efficient approach is to use the Rational Root Theorem. The Rational Root Theorem states that if a rational number p/q is a zero of a polynomial equation with integer coefficients, then p must be a factor of the constant term (in this case, 81 for P(x) and -25 for H(x)), and q must be a factor of the leading coefficient (1 for P(x) and 1 for H(x)).
Let's apply the Rational Root Theorem to each function:
For P(x) = x^4 - 7x^3 - 27x^2 + 63x + 81:
- The potential rational roots are the factors of the constant term (81) divided by the factors of the leading coefficient (1): ±1, ±3, ±9, ±27, ±81.
- Checking these potential roots, we find that none of them are zeros of P(x).
For H(x) = x^3 - x^2 + 25x - 25:
- The potential rational roots are the factors of the constant term (-25) divided by the factors of the leading coefficient (1): ±1, ±5, ±25.
- Checking these potential roots, we find that none of them are zeros of H(x).
Therefore, based on both the examination of coefficients and the application of the Rational Root Theorem, neither function P(x) nor H(x) has zeros that are complex conjugates.