Suppose that a polynomial function of degree 5 with rational coefficients has 6, −2+4i, 4−sqrt2 as zeros. Find the other zeros.
Thank you!
6
-2+4i needs -2-4i
that is three, we need two more but your last one is real, therefore I suspect a typo
perhaps it should be
4 - sqrt-2 which is 4 -i sqrt 2
then we would have
4 - i sqrt2 and 4 + i sqrt 2
To find the other zeros of the polynomial function, we need to use the conjugate pairs theorem. This theorem states that if a polynomial with real coefficients has a complex root, then its conjugate is also a root.
Given that the zeros are 6, -2+4i, and 4-sqrt(2), we know that the complex roots must come in conjugate pairs. We can split -2+4i into two parts: -2 (real part) and 4i (imaginary part).
So, the other complex root in the conjugate pair of -2+4i is -2-4i.
Therefore, the zeros of the polynomial function are 6, -2+4i, -2-4i, 4-sqrt(2), and the other two zeros are the conjugates of these complex roots.
Hence, the other zeros are -2-4i and 4+sqrt(2).
To find the other zeros of the polynomial function, we need to consider the conjugate of the given complex zeros.
For the polynomial to have rational coefficients, complex zeros must occur in conjugate pairs.
Given zeros:
- 6 (a real zero)
- -2+4i (a complex zero)
- 4-√2 (a real zero)
Since -2+4i is a complex zero, its conjugate is -2-4i.
So, the list of zeros now becomes:
- 6 (a real zero)
- -2+4i (a complex zero)
- -2-4i (the conjugate of -2+4i)
- 4-√2 (a real zero)
Now, we can solve for the polynomial function by using the zeros.
Let's call the polynomial function f(x), and the zeros as x1, x2, x3, x4, and x5.
f(x) = (x - x1)(x - x2)(x - x3)(x - x4)(x - x5)
= (x - 6)(x - (-2+4i))(x - (-2-4i))(x - (4-√2))(x - ??)
To find the fifth zero, we can use the property of polynomials. The sum of the zeros of a polynomial function is equal to the opposite of the coefficient of the x to the power of (n-1), where n is the degree of the polynomial.
In this case, the polynomial degree is 5, so the sum of the zeros is -coefficients of x^4.
The sum of the zeros = x1 + x2 + x3 + x4 + x5 = 0
We have already found x1, x2, x3, and x4. So, substituting the values:
0 = -6 + (-2+4i) + (-2-4i) + (4-√2) + x5
Now, simplify the equation:
0 = -6 + (-2+4i) + (-2-4i) + (4-√2) + x5
= -6 - 2 + 4i - 2 - 4i + 4 - √2 + x5
= -6 -2 - 2 + 4 + 4i - 4i - √2 + x5
= -10 + x5
Solving for x5, we get:
x5 = 10
Therefore, the other zero of the polynomial function is 10.