Problem 82 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{3 p^{2}+15 p+12}{p^{2}+4 p+3} \div \frac{18}{6 p+18} $$

Step-by-Step Solution

Answer

\(p+4\).

Step 1: Factorize the Numerator of the First Fraction

Factor the numerator \(\frac{3p^2 + 15p + 12}{p^2 + 4p + 3}\). Find the roots by factoring: \(\frac{3(p^2 + 5p + 4)}{p^2 + 4p + 3} \). Further breaking it down: \(\frac{3(p+3)(p+4)}{p^2 + 4p + 3} \).

Step 2: Factorize the Denominator of the First Fraction

Factor the denominator: \(\frac{3(p+3)(p+4)}{(p+3)(p+1)} \). The first fraction now is: \(\frac{3(p+4)}{(p+1)} \).

Step 3: Factorize the Second Fraction

Rewrite the second fraction \(\frac{18}{6p+18}\) and factor it. Simplify the denominator: \(\frac{18}{6(p+3)}\) becomes: \(\frac{3}{p+3}\).

Step 4: Division into Multiplication

Convert the division into multiplication: \(\frac{3(p+4)}{(p+1)} \times \frac{p+3}{3} \).

Step 5: Simplification

Simplify the multiplication: \(\frac{(p+4)(p+3)}{(p+1)} \times \frac{1}{1} = \frac{(p+4)}{1} = (p+4)\).

Key Concepts

factoring polynomials

When working with algebraic fractions, one crucial step is factoring polynomials. Factoring helps to break down complex expressions into simpler, more manageable parts. For instance, in the given exercise, we begin by factoring the numerator 3p^2 + 15p + 12. Identifying common factors or using methods like the quadratic formula can simplify this process. In this case, we find that 3(p+3)(p+4) is the factorized form. This step reveals the roots of the polynomial, making it easier to simplify the fraction.

rational expressions

A rational expression is simply a fraction where the numerator and/or the denominator are polynomials. Understanding rational expressions is essential for simplifying algebraic fractions. Just like with regular fractions, you can add, subtract, multiply, and divide rational expressions by following specific rules. In the given problem, both the initial fractions are rational expressions: \(\frac{3(p+3)(p+4)}{(p+3)(p+1)}\) and \(\frac{18}{6(p+3)}\). Simplifying rational expressions often involves factoring polynomials and reducing common factors.

multiplication of fractions

To multiply fractions is straightforward: multiply the numerators together and the denominators together. However, when dealing with algebraic fractions, first simplify both fractions if possible. In our exercise, we simplify \(\frac{3(p+4)}{(p+1)}\) and \(\frac{p+3}{3}\) before multiplication: \(\frac{3(p+4)}{(p+1)} \times \frac{p+3}{3}\). Notice how the 3 cancels out in the numerator and denominator, simplifying the expression to \(\frac{(p+4)(p+3)}{(p+1)}\). This reduces the complexity of further calculations.

division of fractions

Dividing fractions involves multiplying by the reciprocal of the divisor. This means flipping the second fraction and then multiplying. In our exercise, we convert the division into multiplication: \(\frac{3(p+4)}{(p+1)} \times \frac{p+3}{3}\). Conversion simplifies the complex operation, turning it into a problem involving the multiplication of fractions. Always remember to simplify both fractions before multiplying for the easiest computation.