Answer :
To simplify the fraction (6x - 3) divided by [tex](x^3 + 3x^2 - 4)[/tex], factor both the numerator and the denominator. The simplified form is [tex](3(2x - 1))\div((x + 1)(x^2 + 2x - 4))[/tex], achieved after dividing the cubic polynomial by its root and factoring the results.
To convert the fraction (6x - 3) divided by [tex](x^3 + 3x^2 - 4)[/tex] 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 [tex]3(2x - 1).[/tex]
Factorize the Denominator:
The denominator is a cubic polynomial, [tex]x^3 + 3x^2 - 4[/tex]. 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 [tex]x^3 + 3x^2 - 4 by x + 1[/tex], leading to the quotient [tex](x^2 + 2x - 4).[/tex]
Thus, [tex]x^3 + 3x^2 - 4[/tex] factors into [tex](x + 1)(x^2 + 2x - 4).[/tex]
Simplify the Fraction:
Combine the factored forms to get: [tex](3(2x - 1))\div((x + 1)(x^2 + 2x - 4)).[/tex]
There are no common factors between the numerator and the denominator that could simplify further.
Thus, the simplified form of the given fraction is [tex](3(2x - 1))\div((x + 1)(x^2 + 2x - 4)).[/tex]
This process involves factoring and synthetic division, commonly taught in high school algebra courses.
