Factor the following polynomial completely, SHOWING ALL STEPS. LaTeX: 6x^3-3x^2+10x-5
Answers
Answer:
6x³ - 3x² + 10x - 5 = (2x - 1)(3x² + 5)
Step-by-step explanation:
First find one root, as this gives a linear factor and factorizing the remaining quadratic is easy. For a tidy factorization, we would hope that there is a rational root.
Suppose then that x = a/b is a root, with a and b relatively prime.
Then 6a³ - 3a²b + 10ab² - 5b³ = 0. ...(*)
This tells us that a | 5 and b | 6, so we have only 16 possible rational roots to try:
- ±1, ±5, ±1/2, ±5/2, ±1/3, ±5/3, ±1/6, ±5/6.
We could try all of these, or we could try to be a bit smarter first...
If a = ±5, then (*) gives 25 | 5b³ ⇒ 5 | b, which is impossible since b | 6. So a = ±1. That reduces the list of possible rational roots to only 8.
Also, if 3 | b, then (*) gives 9 | 6a³ ⇒ 3 | a, which is impossible since a | 5. So b | 2. This reduces the list of candidates to just 4:
- ±1, ±1/2
Putting x = 1 into 6x³ - 3x² + 10x - 5 gives 6 - 3 + 10 - 5 ≠ 0.
Putting in x = -1 gives -6 - 3 - 10 - 5 ≠ 0.
Putting in x = 1/2 gives 6/8 - 3/4 + 10/2 - 5 = 0 ... so x = 1/2 is a root and therefore (2x - 1) is a factor.
Performing the division gives:
- 6x³ - 3x² + 10x - 5 = (2x - 1)(3x² + 5).
As the discriminant of 3x²+5 is negative (it is 0² - 4×3×5), the quadratic factor doesn't factorize any further.