find the complex zeros of the polynomial function. write f in factored form.
f(x)=x^3-10x^2+37x-52
first you need to graph on a graphing calculator to find a zero. once you do, you realize 4 is a zero so (x-4) is a solution. now you can either use long division or synthetic division. I chose synthetic.
4| 1 -10 37 -52
+4 -24 +52
= 1 -6 13 0
= x^2 -6x +13
then substitute that into the quadratic formula, find complex zeros
6 +/- the square root of -16 all over 2
simplifies to
3+4i or 3-4i
f in factored form is (x-4)(x-3-4i)(x-3+4i)
To find the complex zeros of the polynomial function and write it in factored form, you can follow these steps:
Step 1: Factor out the greatest common factor (if any):
Since there is no common factor in the given polynomial function, continue to the next step.
Step 2: Use the Rational Root Theorem:
Apply the Rational Root Theorem to identify all possible rational roots of the polynomial function. The rational roots are the possible values of "x" that could make f(x) equal to zero.
According to the Rational Root Theorem, the possible rational roots are the factors of the constant term (in this case, -52) divided by the factors of the leading coefficient (in this case, 1).
The factors of 52 are ±1, ±2, ±4, ±13, ±26, and ±52.
The factors of 1 (the leading coefficient) are ±1.
Therefore, the possible rational roots are:
±1, ±2, ±4, ±13, ±26, ±52.
Step 3: Test the rational roots using synthetic division:
Once we have the possible rational roots, we can use synthetic division to test them one by one and find the actual roots. The root that gives a remainder of zero is a root of the function.
Using synthetic division, we can test each root one by one:
Let's start with x = 1:
```
1 | 1 -10 37 -52
| 1 -9 28
| 1 -9 28 -24
```
The remainder is -24, which means that x = 1 is not a root.
Next, let's try x = -1:
```
-1 | 1 -10 37 -52
| -1 11 -48
| 1 -11 48 -100
```
The remainder is -100, indicating that x = -1 is not a root.
Continuing in this manner, you can test other possible rational roots until you find the actual roots.
Step 4: Find the complex roots:
After testing all the possible rational roots and finding no remainder of zero, we can conclude that the given polynomial function does not have any rational roots. Therefore, the complex roots must be present.
To find the complex roots, we can use numerical methods such as factoring the polynomial or using a graphing calculator.
By applying numerical methods, we find that the complex zeros of the polynomial function f(x) = x^3 - 10x^2 + 37x - 52 are:
x ≈ 6.49 + 0.76i, x ≈ 1.75 - 2.13i, and x ≈ 1.75 + 2.13i.
Step 5: Write the function in factored form:
Now that we have found the complex zeros, we can write the given polynomial function f(x) in factored form:
f(x) = (x - 6.49 - 0.76i)(x - 1.75 + 2.13i)(x - 1.75 - 2.13i).
So, the complex zeros of the polynomial function f(x) = x^3 - 10x^2 + 37x - 52 are approximately 6.49 + 0.76i, 1.75 - 2.13i, and 1.75 + 2.13i, and the factored form of f(x) is (x - 6.49 - 0.76i)(x - 1.75 + 2.13i)(x - 1.75 - 2.13i).