Find the real and complex zeros of the following function.
f(x)=x^3-3x^2+25x+29
x = -1 is one root, as you can easily verify by inspection. That means that (x +1) is a factor of the polynomial.
Divide x^3-3x^2+25x+29 by x+1 (using polynomial long division) to get a second order polynomial, from which you can get the other roots easily, using the quadratic formula. They will be complex in this case.
To find the real and complex zeros of the function f(x) = x^3 - 3x^2 + 25x + 29, we can use the Rational Root Theorem and synthetic division.
Step 1: List all possible rational roots.
According to the Rational Root Theorem, the possible rational roots of the function are the factors of the constant term (29) divided by the factors of the leading coefficient (1).
The factors of 29 are ±1, ±29, and the factors of 1 are ±1. So, the possible rational roots are ±1 and ±29.
Step 2: Use synthetic division to test the possible roots.
We will test each possible root by applying synthetic division. Starting with x = 1:
1 | 1 -3 25 29
- 1 -2 23
-----------------
1 -2 23 52
Since the last term is not zero, x = 1 is not a root of the function.
Next, we try x = -1:
-1 | 1 -3 25 29
-1 4 -29
----------------
1 -4 21 0
The remainder is zero; therefore, x = -1 is a root of the function.
Now, we try x = 29:
29 | 1 -3 25 29
29 780 16470
----------------------
1 26 805 16499
The remainder is not zero; therefore, x = 29 is not a root of the function.
Lastly, we try x = -29:
-29 | 1 -3 25 29
-29 870 -4640
------------------------
1 -32 895 -4611
The remainder is not zero; therefore, x = -29 is not a root of the function.
Step 3: Determine the roots.
From the synthetic division, we found that x = -1 is a root of the function. Therefore, (x + 1) is a factor of f(x).
To find the remaining factor, divide f(x) by (x + 1) using polynomial long division:
x^2 - 4x + 21
Now, we need to find the roots of the quadratic equation x^2 - 4x + 21 = 0. We can use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -4, and c = 21. Plugging these values into the formula:
x = (-(-4) ± √((-4)^2 - 4(1)(21))) / (2 * 1)
= (4 ± √(16 - 84)) / 2
= (4 ± √(-68)) / 2
= (4 ± 2√(-17)) / 2
= 2 ± √(-17)
Since there is a square root of a negative number, the solutions are complex or imaginary numbers. We can express the complex zeros as:
x = 2 + √(-17)
x = 2 - √(-17)
Therefore, the real zero of the function is x = -1, and the complex zeros are x = 2 + √(-17) and x = 2 - √(-17).