Problem 54 - College Algebra

Question

Perform the given operations and simplify. $$ \frac{x^{2}+x-6}{x^{2}-2 x-3} \cdot \frac{2 x^{2}-3 x-9}{x^{2}-x-2} \div \frac{10 x^{2}+27 x+18}{x^{2}+2 x+1} $$

Step-by-Step Solution

Answer

\(\frac{(x + 3)(2x + 3)(x + 1)}{(5x + 3)(2x + 6)}\) is the simplified result.

Step 1: Factor the Numerators and Denominators

First, factor all polynomials in the given expression. This involves finding values of variables that satisfy each equation when set to zero, then rewriting them as products of binomials.The factors are:- Numerator of the first fraction: \(x^2 + x - 6 = (x + 3)(x - 2)\)- Denominator of the first fraction: \(x^2 - 2x - 3 = (x - 3)(x + 1)\)- Numerator of the second fraction: \(2x^2 - 3x - 9 = (2x + 3)(x - 3)\)- Denominator of the second fraction: \(x^2 - x - 2 = (x - 2)(x + 1)\)- Numerator of the third fraction: \(10x^2 + 27x + 18 = (5x + 3)(2x + 6)\)- Denominator of the third fraction: \(x^2 + 2x + 1 = (x + 1)^2\)

Step 2: Rewrite the Expression

Now that the polynomials are factored, rewrite the original expression using their factored forms:\[\frac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \cdot \frac{(2x + 3)(x - 3)}{(x - 2)(x + 1)} \div \frac{(5x + 3)(2x + 6)}{(x + 1)^2}\]

Step 3: Convert Division to Multiplication

To convert the division of fractions into multiplication, multiply by the reciprocal of the third fraction:\[\frac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \cdot \frac{(2x + 3)(x - 3)}{(x - 2)(x + 1)} \cdot \frac{(x + 1)^2}{(5x + 3)(2x + 6)}\]

Step 4: Simplify by Canceling Common Factors

Cancel terms in the numerators and denominators that appear on both sides of the fractions. The common factors are:- \((x - 2)\) from the first numerator and second denominator.- \((x + 1)\) from both the first denominator, second denominator, and one from the third numerator.- \((x - 3)\) appears in the first denominator and second numerator.This leaves us with:\[\frac{(x + 3)(2x + 3)(x + 1)}{(5x + 3)(2x + 6)}\]

Step 5: Final Check

Ensure that everything is canceled correctly and that no further simplification is needed. The expression is now fully simplified.

Key Concepts

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler components called factors. These factors are polynomials of lower degree. Factoring is crucial in algebra because it simplifies expressions and equations, making them easier to work with. To factor a polynomial,

• identify values that make the polynomial equal zero — these are the roots.
• Rewrite the polynomial as a product of binomials using these roots.

For example, consider the polynomial \(x^2 + x - 6\). By finding values for \(x\) that satisfy the equation \(x^2 + x - 6 = 0\), we discover that it can be factored into \((x + 3)(x - 2)\). Thus, \(x = -3\) and \(x = 2\) are the roots. Factoring is an essential first step when dealing with algebraic fractions as it sets the stage for further operations like simplification or multiplication.

Fraction Multiplication

Multiplying fractions involves multiplying the numerators together and the denominators together. This rule applies straightforwardly to algebraic fractions as well. Consider two fractions:

• \(\frac{a}{b}\)
• \(\frac{c}{d}\)

Their product is \(\frac{a \cdot c}{b \cdot d}\). In an algebraic context with expressions like \(\frac{(x + 3)(x - 2)}{(x - 3)(x + 1)} \cdot \frac{(2x + 3)(x - 3)}{(x - 2)(x + 1)}\), you multiply all numerators and all denominators:

• Numerator: \((x + 3)(x - 2)(2x + 3)(x - 3)\)
• Denominator: \((x - 3)(x + 1)(x - 2)(x + 1)\)

The next step involves simplifying the result by canceling common factors, but this basic operation is crucial to correctly handling such expressions.

Simplifying Fractions

Simplifying fractions is all about reducing the expression to its most compact form. Whenever possible, the goal is to eliminate terms from both the numerator and the denominator that can be canceled out. Steps include:

• Factor both the numerator and denominator completely.
• Identify common factors in the numerator and denominator.
• Cancel these common factors out to simplify the fraction.

For example, in the expression \(\frac{(x + 3)(2x + 3)(x + 1)}{(5x + 3)(2x + 6)}\), if \((x + 1)\) appears both in the numerator and in one of the terms of the denominator, it can be canceled, leading to a simplified expression. The fewer terms you have, the easier the expression is to work with, whether you're solving equations or merely simplifying.

Reciprocals in Division

In algebra, division of fractions can be transformed into multiplication by utilizing reciprocals. A reciprocal of a fraction \(\frac{a}{b}\) is simply \(\frac{b}{a}\). This simplifies complex operations. When dividing two fractions, such as \(\frac{A}{B} \div \frac{C}{D}\), perform the following:

• Convert the division sign into a multiplication sign.
• Find the reciprocal of the divisor \(\frac{C}{D}\), which is \(\frac{D}{C}\).
• Proceed with the multiplication: \(\frac{A}{B} \cdot \frac{D}{C}\).

This conversion transforms a potentially tedious division problem into a straightforward multiplication problem, like in our exercise expression after step 3, which converts the original division into a product, making further simplification more manageable.