Problem 65 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{5 a^{2} y+3 a^{2}}{2 x^{3}+5 x^{2}} \div \frac{10 a y+6 a}{6 x^{3}+15 x^{2}}$$

Step-by-Step Solution

Answer

The result of the division is \( \frac{4}{6x+15} = \frac{(5y+3)(6x+15)}{(2 x+5)(10 y+6)} \)

Step 1: Write the Division As Multiplication

We know that \( a \div b = a \times \frac{1}{b} \). Therefore the initial problem converts to:\( \frac{5 a^{2} y+3 a^{2}}{2 x^{3}+5 x^{2}} \times \frac{6 x^{3}+15 x^{2}}{10 a y+6 a} \)

Step 2: Simplify the fractions

We can simplify each term inside the fractions:\( \frac{a^{2}(5y+3)}{x^{2}(2 x+5)} \times \frac{x^{2}(6x+15)}{a(10 y+6)} \)

Step 3: Remove Common Factors

Notice that \( a^2 \) cancels out with \( a \) from the fraction on the right, and \( x^2 \) cancels out with \( x^{2} \) from the fraction on the left. The result:\( \frac{5y+3}{2 x+5} \times \frac{6x+15}{10 y+6} \)

Step 4: Simplify the Multiplication

Multiply the two simplified fractions to find the solution:\( \frac4{6x+15} = \frac{(5y+3)(6x+15)}{(2 x+5)(10 y+6)} \)

Key Concepts

Division of Rational Expressions

Dividing rational expressions involves an important concept: turning division into multiplication. When you see a division of fractions, such as \( \frac{A}{B} \div \frac{C}{D} \), you can rewrite it as a multiplication by the reciprocal of the second fraction. Thus, it becomes \( \frac{A}{B} \times \frac{D}{C} \). This step is crucial as it simplifies the process of dealing with fractions by handling it with multiplication.

• First, rewrite the division as a multiplication.
• Find the reciprocal of the divisor (flip the second fraction).
• Multiply the numerators together and the denominators together.

In our initial exercise, we turned \( \frac{5 a^{2} y+3 a^{2}}{2 x^{3}+5 x^{2}} \div \frac{10 a y+6 a}{6 x^{3}+15 x^{2}} \) into \( \frac{5 a^{2} y+3 a^{2}}{2 x^{3}+5 x^{2}} \times \frac{6 x^{3}+15 x^{2}}{10 a y+6 a} \). This conversion makes everything much simpler moving forward.

Simplifying Rational Expressions

Simplifying rational expressions is akin to reducing fractions. The concept here is to identify and cancel out common factors in both the numerator and the denominator.
Just as you might simplify \( \frac{8}{12} \) to \( \frac{2}{3} \) by canceling the common factor of 4, we do the same with algebraic expressions.

• Factor each part of the expression to its simplest terms.
• Look for common factors in both numerator and denominator.
• Cancel as many common factors as possible.

In the worked exercise, we simplified \( \frac{5 a^{2} y+3 a^{2}}{2 x^{3}+5 x^{2}} \) to \( \frac{a^{2}(5y+3)}{x^{2}(2 x+5)} \) and \( \frac{6 x^{3}+15 x^{2}}{10 a y+6 a} \) to \( \frac{x^{2}(6x+15)}{a(10 y+6)} \). By doing so, common factors like \( a^2 \) and \( x^2 \) were easily canceled out.

Multiplication of Rational Expressions

Multiplying rational expressions is straightforward once they are simplified. You simply multiply the numerators together and the denominators together.

Remember the key points:

• Simplify each expression before multiplication to make things easier.
• Multiply straight across, combining numerators and denominators.
• Check to see if the resultant expression can be further simplified.

In our example, the final multiplication was \( \frac{5y+3}{2 x+5} \times \frac{6x+15}{10 y+6} \), resulting in \( \frac{(5y+3)(6x+15)}{(2 x+5)(10 y+6)} \). It is essential to verify if further simplification is possible, as sometimes additional common factors become apparent after multiplication.