Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{\frac{5 x^{2}-7 x-6}{2 x^{2}+3 x-5}}{\frac{15 x^{2}+14 x+3}{2 x^{2}+13 x+20}} $$

Short answer

\( \frac{(x - 2)(x + 4)}{(x - 1)(3x + 1)} \)

Step-by-step solution

- Simplify the complex fraction

Rewrite the complex fraction as a division of two fractions: \[ \frac{\frac{5x^2 - 7x - 6}{2x^2 + 3x - 5}}{\frac{15x^2 + 14x + 3}{2x^2 + 13x + 20}} = \frac{5x^2 - 7x - 6}{2x^2 + 3x - 5} \times \frac{2x^2 + 13x + 20}{15x^2 + 14x + 3} \]

- Factor the polynomials in the numerator and the denominator

Factor each polynomial in the numerator and the denominator: \(5x^2 - 7x - 6 = (5x + 3)(x - 2)\) \(2x^2 + 3x - 5 = (2x + 5)(x - 1)\) \(15x^2 + 14x + 3 = (3x + 1)(5x + 3)\) \(2x^2 + 13x + 20 = (2x + 5)(x + 4)\)

- Substitute the factored expressions

Replace the original polynomials with their factored forms: \[ \frac{(5x + 3)(x - 2)}{(2x + 5)(x - 1)} \times \frac{(2x + 5)(x + 4)}{(3x + 1)(5x + 3)} \]

- Cancel out the common factors

Simplify the expression by cancelling the common factors: \[ \frac{(\cancel{5x + 3})(x - 2)}{(\cancel{2x + 5})(x - 1)} \times \frac{(\cancel{2x + 5})(x + 4)}{(3x + 1)(\cancel{5x + 3})} \] This simplifies to: \[ \frac{(x - 2)(x + 4)}{(x - 1)(3x + 1)} \]

Key concepts

Factoring Polynomials

To start with simplifying the given complex fraction, it’s crucial to factor each polynomial in the numerator and the denominator. Factoring is the process of breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give the original polynomial. For instance:
- The polynomial \(5x^2 - 7x - 6\) can be factored as \((5x + 3)(x - 2)\).
- Similarly, \(2x^2 + 3x - 5\) becomes \((2x + 5)(x - 1)\).
- For the denominator fractions, \(15x^2 + 14x + 3\) factors into \((3x + 1)(5x + 3)\), and \(2x^2 + 13x + 20\) into \((2x + 5)(x + 4)\).
This step ensures we can have simpler units to work with in the next steps. Remember, each factorization is like uncovering the 'simpler building blocks' of each polynomial.

Simplifying Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both, contain fractions themselves. To simplify a complex fraction, rewrite it as a division of two simpler fractions. For example:
First, we take the given expression \(\frac{\frac{5x^2 - 7x - 6}{2x^2 + 3x - 5}}{\frac{15x^2 + 14x + 3}{2x^2 + 13x + 20}}\) and rewrite it as: \begin{align*} \frac{5x^2 - 7x - 6}{2x^2 + 3x - 5} \times \frac{2x^2 + 13x + 20}{15x^2 + 14x + 3} \ \text{This rephrasing transforms the complex fraction into a multiplication problem of two rational expressions.} Therefore, turning the complex fraction into simpler pieces is crucial before moving to the next stage.

Cancelling Common Factors

Once we have factored polynomials and rewritten the complex fraction as a product of simpler fractions, it's time to cancel common factors. This simply means reducing the expression by eliminating identical factors from the numerator and the denominator. Here’s how that works:
Write the simplified form by substituting factored expressions:\(\frac{(5x + 3)(x - 2)}{(2x + 5)(x - 1)} \times \frac{(2x + 5)(x + 4)}{(3x + 1)(5x + 3)}\).
In this expression, both \((5x + 3)\) and \((2x + 5)\) occur in the numerator and the denominator. We can cancel these out because any non-zero number divided by itself equals one. This yields: \(\frac{(x - 2)(x + 4)}{(x - 1)(3x + 1)}\).
Always look for and cancel these common factors to simplify the complex fraction efficiently.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division of algebraic expressions. In the context of rational expressions, multiplication and division are particularly relevant. When dividing two fractions, you multiply by the reciprocal of the second fraction. For multiplication, you directly multiply the numerators together and the denominators together before factoring or simplifying if possible.
In the exercise, starting with the rewritten form \(\frac{5x^2 - 7x - 6}{2x^2 + 3x - 5} \times \frac{2x^2 + 13x + 20}{15x^2 + 14x + 3}\): 1. Multiply the numerators together: \((5x^2 - 7x - 6) \times (2x^2 + 13x + 20)\) 2. Multiply the denominators together: \((2x^2 + 3x - 5) \times (15x^2 + 14x + 3)\)
This helps in reducing complex algebraic operations into simpler steps that you solve easily.