information is given about a polynomial f(x)whose coefficients are real numbers. Find the remaining zeros of f.
degree:5, zeros: -6, 6-i
please help and show all work.
From the given zeros, factors would be
(x+6) , (x-6-i) and (x-6 + i)
the complex factors simplify to x^2 -12x +37
(complex numbers always appear in conjugate pair form)
but that would only produce a cubic, so there has to be multiplicity of the above zeros.
two options :
f(x) = (x+6)^3 (x^2 - 12x + 37)
or
f(x) = (x+6)(x^2 - 12x + 37)^2
To find the remaining zeros of the polynomial f(x), we will make use of the fact that the coefficients of the polynomial are real numbers.
Given:
Degree of the polynomial: 5
Zeros of the polynomial: -6, 6-i
Since the coefficients are real numbers, the complex conjugate of 6-i will also be a zero of the polynomial. The complex conjugate of 6-i is 6+i.
So, the zeros of the polynomial are -6, 6-i, and 6+i.
The general form of a polynomial with degree 5 is:
f(x) = a(x - r)(x - s)(x - t)(x - u)(x - v)
Where a is the leading coefficient, and r, s, t, u, and v are the zeros.
Now, we substitute the given zeros into the polynomial equation:
f(x) = a(x + 6)(x - (6-i))(x - (6+i))(x - r)(x - s)
Expanding the equation gives:
f(x) = a(x + 6)(x^2 - 12x + 37)(x - r)(x - s)
Since f(x) is a polynomial of degree 5, there must be two more zeros, r and s.
We don't have enough information to determine the values of r and s since no further information has been given.
Therefore, the remaining zeros of the polynomial f(x) are r and s, but their exact values cannot be determined without additional information.
To find the remaining zeros of a polynomial, we need to consider the information given about the polynomial and use it to determine the remaining zeros. In this case, the degree of the polynomial is 5, and we are given two zeros: -6 and 6-i.
First, let's analyze the given zero -6. If -6 is a zero of the polynomial, then (x + 6) must be a factor of the polynomial. So, one of the factors of the polynomial is (x + 6).
Next, let's consider the given zero 6-i. Since the coefficients of the polynomial are real numbers, complex zeros always occur in conjugate pairs. This means that if 6-i is a zero, then 6+i must also be a zero. If (x - 6 + i) is a factor of the polynomial, then (x - 6 - i) must also be a factor. So, the factors corresponding to the complex zeros are (x - 6 + i) and (x - 6 - i).
Now to find the remaining zeros, we can divide the polynomial by the factors we have found to reduce the degree of the polynomial.
The polynomial can be represented as:
f(x) = (x + 6)(x - 6 + i)(x - 6 - i)g(x)
where g(x) represents the quotient polynomial obtained after dividing by the known factors.
Since the polynomial has a degree of 5 and we know three of its factors, the degree of g(x) must be 2.
Hence, set up the polynomial division and divide by the known factors (x + 6), (x - 6 + i), and (x - 6 - i) to find the quotient polynomial g(x).
Once the quotient polynomial g(x) is found, you can solve for the remaining two zeros of g(x) to obtain the remaining zeros of the original polynomial f(x).