Problem 56 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{3 a+3 b}{a^{2}-4 a-5} \div \frac{9 a+9 b}{a^{2}-3 a-10} $$

Step-by-Step Solution

Answer

Based on the given step-by-step solution, provide a short answer. Short Answer: The simplified result of the given expression is $\frac{a + 2}{3(a + 1)}$.

Step 1: Rewrite division as multiplication

To rewrite the given expression, we will invert the second fraction and change the division operation into multiplication. $$ \frac{3a + 3b}{a^2 - 4a - 5} \div \frac{9a + 9b}{a^2 - 3a - 10} = \frac{3a + 3b}{a^2 - 4a - 5} \times \frac{a^2 - 3a - 10}{9a + 9b} $$

Step 2: Factorize the numerators and denominators

Next, we will factorize the numerators and denominators to look for any terms that can be simplified. $$ \frac{3(a + b)}{(a - 5)(a + 1)} \times \frac{(a - 5)(a + 2)}{9(a + b)} $$

Step 3: Multiply the numerators and denominators

Now, we will multiply the numerators and denominators while simplifying the result by canceling out any common terms. $$ \frac{3(a + b)}{(a - 5)(a + 1)} \times \frac{(a - 5)(a + 2)}{9(a + b)} = \frac{3(a + b)(a - 5)(a + 2)}{9(a + b)(a - 5)(a + 1)} $$ After canceling out the common terms \((a + b)\), \((a - 5)\), and 3: $$ = \frac{(a + 2)}{3(a + 1)} $$ The final result is: $$ \frac{a + 2}{3(a + 1)} $$

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing expressions to their simplest form, making them easier to understand and work with. For instance, the expression \(3a + 3b\) can be simplified by factoring out the common factor of 3, yielding \(3(a + b)\). Furthermore, simplification often includes canceling out terms that appear in both the numerator and the denominator of a fraction.

When simplifying, always look for:

• Common factors that can be factored out.
• Terms that can be combined or canceled.
• Any potential to use algebraic identities to simplify.

Through simplification, complex expressions become more manageable, which is particularly useful in solving equations, graphing functions, or working with algebraic fractions as in the exercise provided.

Factoring Polynomials

Factoring polynomials is a process of breaking down a polynomial into a product of its factors. Factors of polynomials are simpler expressions, which when multiplied together give back the original polynomial. In our exercise, the denominators \(a^2 - 4a - 5\) and \(a^2 - 3a - 10\) are factored into \(a - 5)(a + 1)\) and \(a - 5)(a + 2)\), respectively.

Effective factoring methods include:

• Finding common factors for all terms.
• Applying the difference of squares formula, if applicable.
• Using the sum or difference of cubes formulas.
• Employing the quadratic formula in suitable situations.

Mastering factoring polynomials is crucial for solving polynomial equations and simplifying algebraic fractions.

Multiplying and Dividing Fractions

The process of multiplying and dividing algebraic fractions follows the same basic principles as multiplying and dividing numerical fractions. To multiply fractions, simply multiply the numerators together and the denominators together. On the other hand, division requires us to multiply by the reciprocal of the divisor fraction.

For division, the steps include:

• Writing the division of fractions as the multiplication of the first fraction by the reciprocal of the second fraction.
• Before multiplying, look for opportunities to simplify, which may involve factoring polynomials as shown in the exercise.
• Multiply the simplified numerators and denominators.

Always remember to reduce the fractions to their simplest form by canceling any common factors. This will also save you time and reduce the complexity of your calculations.