find the complex zeros of the polynomial function. write f in factored form. use complex zeros to write f in factored form
f(x)= x^3-6x^2+21x-26
What do you mean by "complex zeroes"? You must mean the complex values of x that result in a zero for f(x).
Inspection indicates that x = 2 is a root, so (x-2) is a factor. Divide f(x) by x-2 for the remaining quadratic factor, which should be x^2 -4x +13.
That will have two complex roots. Find them.
To find the complex zeros of the polynomial function f(x) = x^3 - 6x^2 + 21x - 26, we can use the Rational Root Theorem and synthetic division.
The Rational Root Theorem states that any rational zero (if it exists) can be expressed as a ratio of a factor of the constant term (in this case, -26) to a factor of the leading coefficient (in this case, 1). The factors of the constant term -26 are ±1, ±2, ±13 and ±26, while the factors of the leading coefficient 1 are ±1. Therefore, the possible rational zeros are ±1, ±2, ±13, and ±26.
To test these possible rational zeros, we perform synthetic division on f(x) with each value:
1 | 1 -6 21 -26
| 1 -5 16
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| 1 -5 16 -10
Since the remainder is not zero, 1 is not a zero of f(x).
-1 | 1 -6 21 -26
| -1 7 -28
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| 1 -7 28 -54
Since the remainder is not zero, -1 is not a zero of f(x).
2 | 1 -6 21 -26
| 2 -8 26
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| 1 -4 13 0
Since the remainder is zero, 2 is a zero of f(x).
Using synthetic division with 2 as the zero, we have:
(x - 2) | (x^3 - 6x^2 + 21x - 26)
(x^2 - 4x + 13)
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(x - 2)(x^2 - 4x + 13)
So, we can write f(x) in factored form as f(x) = (x - 2)(x^2 - 4x + 13).
Therefore, the complex zeros of f(x) are x = 2 + i√3 and x = 2 - i√3.