To simplify the fraction (6x - 3) divided by \((x^3 + 3x^2 - 4)\), factor both the numerator and the denominator. The simplified form is \((3(2x - 1))\div((x + 1)(x^2 + 2x - 4))\), achieved after dividing the cubic polynomial by its root and factoring the results.
To convert the fraction (6x - 3) divided by \((x^3 + 3x^2 - 4)\) into its simplest form, we need to factorize both the numerator and the denominator and then simplify.
Step-by-Step Explanation:
Factorize the Numerator:
The numerator is 6x - 3. This can be factored as \(3(2x - 1).\)
Factorize the Denominator:
The denominator is a cubic polynomial, \(x^3 + 3x^2 - 4\). To factorize it, we can look for potential roots using the Rational Root Theorem. Testing possible roots quickly shows that x = -1 is a root.
After confirming that x = -1 is a root, we can use synthetic division or polynomial division to divide \(x^3 + 3x^2 - 4 by x + 1\), leading to the quotient \((x^2 + 2x - 4).\)
Thus, \(x^3 + 3x^2 - 4\) factors into \((x + 1)(x^2 + 2x - 4).\)
Simplify the Fraction:
Combine the factored forms to get: \((3(2x - 1))\div((x + 1)(x^2 + 2x - 4)).\)
There are no common factors between the numerator and the denominator that could simplify further.
Thus, the simplified form of the given fraction is \((3(2x - 1))\div((x + 1)(x^2 + 2x - 4)).\)
This process involves factoring and synthetic division, commonly taught in high school algebra courses.