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.
disregard, didn't mean to post again, accidently hit button.
that gives three zeros, not five
-6
6-i
6+i complex conjugate of 6-i
sorry, I do not see two more from what you wrote.
see
http://www.jiskha.com/display.cgi?id=1353700793
To find the remaining zeros of the polynomial f(x), we know that the degree of the polynomial is 5, and we are given two zeros: -6 and 6-i.
Since the coefficients are real numbers, we know that complex zeros occur in conjugate pairs. Therefore, if 6-i is a zero, its conjugate 6+i must also be a zero.
So, to find the remaining zeros, we need to factorize the polynomial. Since the degree of the polynomial is 5 and we have 3 zeros (including their conjugates), we can set up a factorization using the Remainder Theorem.
The Remainder Theorem tells us that if a polynomial f(x) is divided by x - a, the remainder is f(a). So, if x - (-6) or x + 6 is a factor, then f(-6) should be equal to zero.
Let's set up the factorization:
(x + 6)(x - 6)(x - (6 - i))(x - (6 + i))(ax + b) = 0
We have a 5th-degree polynomial, so there's one more factor, (ax + b), that represents the remaining factor. But since we don't have enough information to determine the exact value of the last factor, we will leave it as (ax + b).
By using the zero product property, we know that if a product of factors is equal to zero, then at least one of the factors must be zero. Therefore:
(x + 6)(x - 6)(x - (6 - i))(x - (6 + i))(ax + b) = 0
Expanding and simplifying:
(x^2 - 36)(x^2 - 12x + 37)(ax + b) = 0
To find the remaining zeros, we need to find the values of x that make each factor equal to zero.
1. (x + 6) = 0
This gives us x = -6 as one zero.
2. (x - 6) = 0
This gives us x = 6 as another zero.
3. (x - (6 - i)) = 0
Simplifying, we get x = 6 - i as a third zero.
4. (x - (6 + i)) = 0
Simplifying, we get x = 6 + i as the fourth zero.
Therefore, the remaining zeros of the polynomial f(x) are x = -6, 6, 6 - i, and 6 + i.