Problem 31 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{a+a^{2}}{15 a^{2}+11 a+2} \cdot \frac{1-a}{1-a^{2}}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{a}{(3a + 2)(5a + 1)}\).

Step 1: Factor the Expressions

First, we must factor both numerators and denominators.**Numerator of the first fraction:** \(a + a^2 = a(1 + a)\).**Denominator of the first fraction:**\(15a^2 + 11a + 2\). Factor this into \((3a + 2)(5a + 1)\) using factorization techniques like trial and error or splitting the middle term.**Numerator of the second fraction:**\(1-a\).**Denominator of the second fraction:**\(1 - a^2\) can be expressed as \((1 + a)(1 - a)\) using the difference of squares formula.

Step 2: Set Up the Multiplication

Substitute the factored forms back into the multiplication: \[ \frac{a(1 + a)}{(3a + 2)(5a + 1)} \times \frac{1-a}{(1 + a)(1 - a)} \].

Step 3: Simplify the Expression

Cancel out any common factors in the numerators and denominators before performing the multiplication. Notice that the \((1-a)\) term in the second numerator and the \((1-a)\) term in the second denominator cancel each other out. Similarly, \((1+a)\) from the first numerator and \((1+a)\) from the second denominator cancel.After cancelling, we have:\[ \frac{a}{(3a + 2)(5a + 1)} \].

Step 4: Final Product Simplification

Now, perform the multiplication which now becomes unnecessary since all terms have been cancelled or reduced to simplest form:The result is \(\frac{a}{(3a + 2)(5a + 1)}\) and the expression is already in its simplest form as there are no more factors to cancel.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It involves expressing a polynomial as a product of its simpler factors. This process is crucial because it allows us to simplify expressions and solve polynomial equations. To factor a polynomial, we look for common factors and patterns, such as:

• Common monomial factoring
• The difference of squares
• The sum or difference of cubes
• Trinomials and special products

In the given exercise, the numerators and denominators need to be factored. For example, consider the second denominator, which is a difference of squares: \(1 - a^2 = (1 + a)(1 - a)\). Identifying these patterns makes it possible to simplify complex expressions. By breaking down polynomials into their factors, we can see which components can be cancelled or further simplified in a rational expression.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Just like numeric fractions, rational expressions can be simplified, multiplied, or divided. Before manipulating rational expressions, it is essential to factor them completely.

• Identify and cancel common factors in the numerator and denominator.
• Ensure that the expression does not have any terms that cause undefined values, such as division by zero.

In our problem, once we factor both the fractions, we set up the expression to see which factors can cancel each other out. For example, both \(1-a\) and \(1+a\) are factored and canceled, making the expression easier to handle. This simplification is analogous to reducing a numeric fraction by dividing out the greatest common divisor.

Multiplication and Division of Fractions

Understanding how to multiply and divide fractions is crucial when dealing with rational expressions. The process for multiplying fractions involves:

• Multiplying the numerators together to get a new numerator.
• Multiplying the denominators together to get a new denominator.
• Simplifying the resultant fraction if possible.

For dividing fractions, remember to multiply by the reciprocal of the divisor. This means flipping the second fraction and then proceeding with multiplication:

• \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

In the exercise, the fractions are factored and then multiplied, which involves simplifying the result by cancelling common factors. This reduces the expression to its simplest form \(\frac{a}{(3a + 2)(5a + 1)}\). Mastering multiplication and division of fractions ensures a deeper understanding of simplifying rational expressions efficiently.