Problem 65 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{3 p-1}{8 p} \div \frac{3 p^{2}+14 p-5}{6 p^{2}} $$

Step-by-Step Solution

Answer

\(\frac{3 p}{4 (p + 5)}\)

Step 1 - Rewrite the division as multiplication by the reciprocal

Rewrite the division operation as multiplication by the reciprocal of the second fraction. This means turning \(\frac{3 p-1}{8 p} \div \frac{3 p^{2}+14 p-5}{6 p^{2}}\) into \(\frac{3 p-1}{8 p} \times \frac{6 p^{2}}{3 p^{2}+14 p-5}\).

Step 2 - Factorize the quadratic expression

Factorize the quadratic expression in the denominator of the second fraction. \(3 p^{2} + 14 p - 5\) factors into \((3 p - 1)(p + 5)\).

Step 3 - Substitute the factored expression

Replace the denominator \(3 p^{2} + 14 p - 5\) with \((3 p - 1)(p + 5)\). The expression becomes \(\frac{3 p-1}{8 p} \times \frac{6 p^{2}}{(3 p - 1)(p + 5)}\).

Step 4 - Simplify the expression

Cancel the common factors in the numerator and denominator. The \(3 p - 1\) from the numerator of the first fraction cancels with \(3 p - 1\) in the denominator of the second fraction. Similarly, one \(p\) from \(6 p^{2}\) cancels with \(p\) in the denominator of the first fraction. This results in \(\frac{6 p}{8 (p + 5)}\).

Step 5 - Simplify further

Simplify the fraction \(\frac{6 p}{8 (p + 5)}\) by dividing the numerator and denominator by their greatest common divisor, which is 2. This results in \(\frac{3 p}{4 (p + 5)}\).

Key Concepts

Simplifying Fractions

To simplify a fraction, you need to reduce it to its simplest form. This means finding the greatest common divisor (GCD) of the numerator and the denominator, and then dividing both by that number. For instance, if you have \frac{6}{8}$$, the GCD of 6 and 8 is 2. Dividing both 6 and 8 by 2, we get \frac{3}{4}$$. Hence, \frac{6}{8}$$ simplifies to \frac{3}{4}$$.

Simplification also involves canceling out common factors in the numerator and denominator. For example, in the expression \frac{6p}{8}$$, we can cancel out a common factor of 2 to get \frac{3p}{4}$$.

Multiplying Rational Expressions

Multiplying rational expressions involves multiplying the numerators together and the denominators together. For example, when given two fractions \frac{a}{b}$$ and \frac{c}{d}$$, their product is \frac{a \times c}{b \times d}$$.

After multiplying, you should simplify the result by finding and canceling common factors between the numerator and denominator.

As illustrated in the exercise, \frac{3p-1}{8p} \times \frac{6p^{2}}{3p^{2}+14p-5}$$ changes to \frac{3p-1}{8p} \times \frac{6p^{2}}{(3p-1)(p+5)}$$. By canceling common factors, the expression became simpler.

Factoring Quadratic Expressions

Factoring quadratic expressions often simplifies solving and simplifying rational expressions. A quadratic expression typically has the form \textbf{ax^2 + bx + c}$$.

To factor it, look for two numbers that multiply to \textbf{ac}$$ and add to \textbf{b}$$. For example, to factor \textbf{3p^2 + 14p - 5}$$, we need numbers that multiply to -15 (3 * -5) and add to 14. These numbers are 15 and -1.

Thus, \textbf{3p^2 + 14p - 5}$$ factors into \textbf{(3p-1)(p+5)}$$, making it easier to handle in rational expressions.

This skill is crucial in simplifying and solving expressions, as seen in the exercise.