for the function g(x)=5x^6-3x^3+x^2-x, list the possible rational zeros of g(x) and then use synthetic division to factor g(x) completely.
To find the possible rational zeros of the function g(x) = 5x^6 - 3x^3 + x^2 - x, we can use the Rational Root Theorem. According to this theorem, the possible rational zeros will have the form p/q, where p is a factor of the constant term (-1 in this case) and q is a factor of the leading coefficient (5 in this case).
Factors of -1: ±1
Factors of 5: ±1, ±5
So the possible rational zeros are: ±1, ±1/5
Now let's use synthetic division to factor g(x) completely:
We'll start with the possible rational zero x = 1.
The coefficients of the function are: 5, 0, -3, 1, -1, 0.
Using synthetic division:
1 │ 5 0 -3 1 -1 0
└───────────────────
5 5 2 3 2 2
The result is the quotient: 5x^5 + 5x^4 + 2x^3 + 3x^2 + 2x + 2.
Now we'll find the next possible rational zero x = -1.
The coefficients of the new quotient are: 5, 5, 2, 3, 2, 2.
Using synthetic division again:
-1 │ 5 5 2 3 2 2
└───────────────────
5 0 2 1 3 -1
The second quotient is: 5x^4 + 2x^2 + x + 3.
We continue this process with the remaining possible rational zeros (-1/5, -5), until we find all the zeros of the function g(x).
By using synthetic division with the remaining possible rational zeros, we find additional factors of the function g(x), until we reach the point where the degree is reduced to 0.
After completing the synthetic division step-by-step with each possible rational zero, if any zeros are found, the function can be factored completely.