Problem 25 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{3 x+6}{5 y} \cdot \frac{x^{2}+4}{x^{2}+10 x+16}\)

Step-by-Step Solution

Answer

\(\frac{3(x^2+4)}{5y(x+8)}\)

Step 1: Factor Numerators and Denominators

The first rational expression is \( \frac{3x+6}{5y} \). The numerator \( 3x+6 \) can be factored to \( 3(x+2) \). The second rational expression is \( \frac{x^2+4}{x^2 + 10x + 16} \). The denominator \( x^2 + 10x + 16 \) can be factored as \( (x+2)(x+8) \). The expression becomes: \[\frac{3(x+2)}{5y} \cdot \frac{x^2+4}{(x+2)(x+8)}\]

Step 2: Cancel Common Factors

Check for common factors in the numerators and denominators. The factor \( (x+2) \) appears in both the numerator of the first expression and the denominator of the second.Cancel the common factor \( (x+2) \):\[\frac{3}{5y} \cdot \frac{x^2+4}{(x+8)}\]

Step 3: Multiply Remaining Expressions

Multiply the remaining numerators and denominators:Numerator: \( 3(x^2 + 4) \)Denominator: \( 5y(x + 8) \)This results in:\[\frac{3(x^2+4)}{5y(x+8)}\]

Step 4: Simplify Final Expression

Check the expression for any further simplifications. The terms \( 3(x^2+4) \) in the numerator and \( 5y(x+8) \) in the denominator do not have any common factors.Therefore, the expression is already in its simplest form:\[\frac{3(x^2+4)}{5y(x+8)}\]

Key Concepts

Factoring Polynomials

When working with rational expressions, the first step is often to factor polynomials. Factoring helps in simplifying expressions, allowing us to see if any terms cancel out. In our exercise, we had the polynomial in the numerator, \(3x + 6\). To factor it, look for common factors. Here, both terms in the polynomial are divisible by 3, which simplifies it to \(3(x + 2)\).

Similarly, consider the denominator \(x^2 + 10x + 16\). We need to factor it into two binomials. Firstly, check if there is a product of two numbers that sums to 10 (the middle term coefficient) and multiplies to 16 (the constant term). These numbers are 2 and 8, so the factorization is \((x+2)(x+8)\).

Factoring is crucial, as it reveals potential cancellation opportunities between the numerators and denominators in rational expressions.

Simplifying Expressions

After factoring, the next step is simplifying the expression. Simplifying involves canceling out common factors found in the numerator and the denominator. In this problem, we discover the common factor \((x+2)\), which appears both in the numerator \(3(x+2)\) and in the denominator \((x+2)(x+8)\).

By canceling out the shared factor \((x+2)\), we transform the expression to \(\frac{3}{5y} \cdot \frac{x^2+4}{(x+8)}\).

Simplification helps make complex expressions easier to handle and paves the way for further operations, like multiplication, with reduced complexity. It's a powerful tool in algebra that helps avoid unnecessary computations.

Multiplying Rational Expressions

Multiplying rational expressions involves multiplying the numerators together and the denominators together. Our simplified expressions \(\frac{3}{5y}\) and \(\frac{x^2+4}{x+8}\) need to be multiplied. For the numerators, you multiply \(3 \cdot (x^2 + 4)\), resulting in \(3(x^2 + 4)\). Similarly, for the denominators, multiply \(5y \cdot (x + 8)\), resulting in \(5y(x + 8)\).

So, the final expression after multiplication is \(\frac{3(x^2+4)}{5y(x+8)}\).

It's essential to check for any further simplifications after multiplying. However, as shown, if there are no common factors left to cancel, the expression is already in its simplest form. Multiplying rational expressions extends the arithmetic of fractions to algebraic expressions, maintaining the relations between quantities while simplifying and reducing errors.