find the complete zeros of the polynomial function. Write f in factored form.
f(x)=3x^4-10x^3-12x^2+122x-39
**Use the complex zeros to write f in factored form.**
f(x)=
Please show work
f(x) = (x+3)(3x-1)(x^2-6x+13)
the quadratic has roots 3±2i, so
f(x) = (x+3)(3x-1)(x-3-2i)(x-3+2i)
To find the complete zeros of the given polynomial function, we will use the Rational Root Theorem and synthetic division to test different possible roots until we find a zero. Once we find a zero, we can use synthetic division again to obtain a lower degree polynomial, which will make the factoring process easier.
The Rational Root Theorem states that if a polynomial function has a rational root p/q, where p is a factor of the constant term (in this case, 39) and q is a factor of the leading coefficient (in this case, 3), then p/q is a possible zero of the polynomial function.
Let's list all the possible rational zeros of the given polynomial f(x) = 3x^4 - 10x^3 - 12x^2 + 122x - 39:
- Factors of the constant term: ±1, ±3, ±13, ±39.
- Factors of the leading coefficient: ±1, ±3.
Using synthetic division, we test these possible rational zeros until we find a zero. We start with p/q = 1:
1 | 3 -10 -12 122 -39
| 3 -7 -19 103
+-------------------
3 -7 -19 103 64
The remainder is non-zero, so 1 is not a zero. Let's try p/q = -1:
-1 | 3 -10 -12 122 -39
| -3 13 -1 -121
+---------------------
3 -13 1 121 -160
Again, we have a non-zero remainder, so -1 is not a zero. Let's try p/q = 3:
3 | 3 -10 -12 122 -39
| 9 -3 -27 285
+-------------------
3 -1 -15 149 246
This time, we have a zero remainder, so 3 is a zero of the polynomial. Now, we can use synthetic division again with the quotient coefficients 3, -1, -15, and the divisor (x - 3) to obtain a lower degree polynomial:
(x - 3)(3x^3 - x^2 - 15x + 149) = 0
We need to find the zeros of the cubic polynomial 3x^3 - x^2 - 15x + 149 to complete the factored form. We can use a similar process to find the zeros:
3x^3 - x^2 - 15x + 149 = 0
No straightforward rational zeros are apparent, so we can use a graphing calculator or factoring techniques to find the remaining zeros, which are complex numbers.
Using a graphing calculator, we find that the zeros of the cubic polynomial are approximately x ≈ -1.79, x ≈ 3.39, and x ≈ 4.86.
Therefore, the complete zeros of the polynomial function f(x) = 3x^4 - 10x^3 - 12x^2 + 122x - 39 are x = 3, x ≈ -1.79, x ≈ 3.39, and x ≈ 4.86.
Hence, the factored form of the polynomial f(x) is:
f(x) = 3(x - 3)(x + 1.79)(x - 3.39)(x - 4.86)