what is the factor of f(x) = 5x3 2x2 -6x-1?
Look for low-hanging fruit. If you meant
5x3+2x2-6x-1
then f(1) = 0, so
f(x) = (x-1)(5x^2+7x-1)
now you can apply the quadratic formula for the rest.
To find the factors of the given polynomial f(x) = 5x^3 - 2x^2 - 6x - 1, we can use synthetic division or factoring by grouping. Here, I will demonstrate factoring by grouping.
Step 1: Group the terms.
Group the terms of the polynomial together, taking common factors from each group if possible:
f(x) = (5x^3 - 2x^2) + (-6x - 1)
Step 2: Take out the common factor.
Factor out the greatest common factor from each group:
f(x) = x^2(5x - 2) - 1(6x + 1)
Step 3: Factor the remaining binomials.
Factor the remaining binomials:
f(x) = x^2(5x - 2) - 1(6x + 1)
= (5x - 2)(x^2 - 1)
So, the factors of the polynomial f(x) = 5x^3 - 2x^2 - 6x - 1 are (5x - 2) and (x^2 - 1).
To find the factors of the polynomial f(x) = 5x^3 + 2x^2 - 6x - 1, we can use the factor theorem. The factor theorem states that if a polynomial f(x) has a factor (x - a), then f(a) = 0.
Let's try to factor f(x) by looking for a potential factor (x - a) where a is a possible root of f(x).
Step 1: Find the potential rational roots
The potential rational roots can be found using the rational root theorem. According to the rational root theorem, the possible rational roots of a polynomial are all the possible ratios of factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is -1 and the leading coefficient is 5.
The factors of the constant term (-1) are ±1, and the factors of the leading coefficient (5) are ±1 and ±5. Therefore, the potential rational roots are ±1/1, ±1/5.
Step 2: Test the potential roots
To determine if any of the potential roots are actual roots of f(x), substitute each potential root back into f(x) and check if the result is equal to zero.
Let's start by testing x = 1:
f(1) = 5(1)^3 + 2(1)^2 - 6(1) - 1
= 5 + 2 - 6 - 1
= 0
Since f(1) = 0, x = 1 is a root and (x - 1) is a factor of f(x).
Step 3: Divide the polynomial by the factor found in Step 2
Now, we can use synthetic division or long division to divide the polynomial f(x) by (x - 1) to find the remaining factor(s).
Using synthetic division, the process is as follows:
1 | 5 2 -6 -1
|___ 5 7 1
| 5 7 1 0
The result of the synthetic division gives us a new polynomial, 5x^2 + 7x + 1, with a remainder of 0.
So, the factored form of the polynomial f(x) = 5x^3 + 2x^2 - 6x - 1 is (x - 1)(5x^2 + 7x + 1).