Find the complex zeros of the polynomial function. Write f in factored form.
f(x)=3x^4-10x^3-12x^2+122x-39
Please show wrok, I do not understand how to do this.
The 39 and 3 tempt me to try roots like 1,3,1/3. Go for the low-hanging fruit first
use synthetic division to find that
f(-3)=0, so
f(x) = (x+3)(3x^3-19x^2+45x-13)
f(1/3) = 0, so
f(x) = (x+3)(3x-1)(x^2-6x+13)
there are no more real roots
To find the complex zeros of the polynomial function and write it in factored form, you can follow these steps:
Step 1: Use the Rational Root Theorem
Start by applying the Rational Root Theorem to identify potential rational roots or zeros. According to the theorem, the possible rational roots of a polynomial with integer coefficients are given by the factors of the constant term (in this case 39) divided by the factors of the leading coefficient (in this case 3).
The factors of 39 are ±1, ±3, ±13, ±39.
The factors of 3 are ±1, ±3.
So, the possible rational roots are ±1, ±3, ±13, ±39.
Step 2: Apply Synthetic Division
Using synthetic division, test each of the potential rational roots obtained in Step 1 until you find a root that produces a remainder of zero. We will start with ±1 for simplicity.
Trying x = 1:
Perform synthetic division as shown below:
1 | 3 -10 -12 122 -39
| 3 -7 -19 103
_____________________
3 -7 -19 103 64
The remainder is 64, so x = 1 is not a root.
Trying x = -1:
Perform synthetic division as shown below:
-1 | 3 -10 -12 122 -39
| -3 13 -1 -121
_____________________
3 -13 1 -121 -160
The remainder is -160, so x = -1 is not a root either.
Continue this process with the other possible rational roots: ±3, ±13, ±39, until you find a root that produces a remainder of zero.
After completing this process, you will find the rational root x = 3.
Step 3: Factor the Polynomial
Now that we have found one rational root, x = 3, the polynomial can be factored by dividing it by (x - 3) using either long division or synthetic division:
(x - 3)(3x^3 + 7x^2 - 46x + 13)
At this point, you have factored the original polynomial into (x - 3)(3x^3 + 7x^2 - 46x + 13).
Step 4: Find the Complex Zeros
The polynomial left inside the parentheses, 3x^3 + 7x^2 - 46x + 13, is a cubic polynomial. To find the remaining complex zeros, you can either continue the process with the Rational Root Theorem or use numerical methods like graphing or using software.
By further factoring or using numerical methods, you can find the complex zeros of the cubic polynomial.
Hence, the complex zeros of the original polynomial function f(x) are x = 3 and the remaining complex zeros from the cubic factor.