Problem 82 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{-2 b^{2}-2 b+4}{8 b^{2}-28 b-16} \div \frac{b^{2}-2 b+1}{2 b^{2}-5 b-3} $$

Step-by-Step Solution

Answer

Question: Divide the rational expressions and simplify the result: $$ \frac{-2b^2-2b+4}{8b^2-28b-16} \div \frac{b^2-2b+1}{2b^2-5b-3} $$ Answer: The simplified result of the division is: $$ \frac{-2(b + 2)(b - 3)}{4(b - 4)(b - 1)} $$

Step 1: Factorize the expressions

Factorize both numerators and denominators of the given rational expressions: $$ \frac{-2b^2-2b+4}{8b^2-28b-16} \div \frac{b^2-2b+1}{2b^2-5b-3} $$ The numerator of the first expression can be factored as: $$ -2b^2 - 2b + 4 = -2(b^2 + b - 2) = -2(b - 1)(b + 2) $$ The denominator of the first expression can be factored as: $$ 8b^2 - 28b - 16 = 4(2b^2 - 7b - 4) = 4(2b + 1)(b - 4) $$ The numerator of the second expression is a perfect square trinomial and can be factored as: $$ b^2 - 2b + 1 = (b - 1)^2 $$ The denominator of the second expression can be factored as: $$ 2b^2 - 5b - 3 = (b - 3)(2b + 1) $$ Rewriting the expressions with their factors: $$ \frac{-2(b - 1)(b + 2)}{4(2b + 1)(b - 4)} \div \frac{(b - 1)^2}{(b - 3)(2b + 1)} $$

Step 2: Rewrite Division as Multiplication

To rewrite the expression as multiplication, we need to find the reciprocal of the second fraction and multiply it with the first fraction: $$ \frac{-2(b - 1)(b + 2)}{4(2b + 1)(b - 4)} \times \frac{(b - 3)(2b + 1)}{(b - 1)^2} $$

Step 3: Simplify the expression

Now, cancel out the common factors from the numerators and the denominators: $$ \frac{-2(b - 1)(b + 2)}{4(2b + 1)(b - 4)} \times \frac{(b - 3)(2b + 1)}{(b - 1)^2} = \frac{-2 \cancel{(b - 1)}(b + 2)}{4 \cancel{(2b + 1)}(b - 4)} \times \frac{(b - 3) \cancel{(2b + 1)}}{\cancel{(b - 1)}(b - 1)} $$ Now, multiply the simplified fractions: $$ \frac{-2(b + 2)(b - 3)}{4(b - 4)(b - 1)} $$ We have reached the simplified expression, and our final answer is: $$ \frac{-2(b + 2)(b - 3)}{4(b - 4)(b - 1)} $$

Key Concepts

Factoring Polynomials

When tackling rational expressions, one of the key steps is factoring polynomials. Factoring involves rewriting a polynomial as a product of its simpler factors that, when multiplied together, give the original polynomial. Consider this akin to breaking down a number into its prime factors.
In our example, we have two polynomials in the numerators and denominators. For instance, the polynomial \(-2b^2-2b+4\) can be factored as \(-2(b-1)(b+2)\), and another, \(b^2-2b+1\), is a perfect square trinomial, which factored becomes \((b-1)^2\).
Understanding the properties of polynomials is crucial to recognize patterns like the difference of squares or perfect square trinomials. Keep practicing! Identifying these patterns will become second nature, making factoring much easier over time.

Simplifying Fractions

The art of simplifying fractions involves making the fraction as simple as possible without changing its value. Simplification often requires canceling out common factors between the numerator and the denominator.
For example, if we have a fraction like \(\frac{-2(b-1)(b+2)}{4(2b+1)(b-4)}\), simplifying involves canceling the common terms in both the numerator and the denominator, such as \((b-1)\) and \((2b+1)\) in our case.
Always factorize first. This will help reveal common factors that can be canceled. Once simplified, it's far easier to work further with these expressions, especially when performing operations like multiplication or division.

Division of Polynomials

Polynomials can also be divided, very much like regular numbers. Division of polynomials becomes straightforward when we use rational expressions. In this process, we often convert division into multiplication by finding the reciprocal of the divisor and then multiplying.
In our exercise, the division \(\frac{\text{Expression 1}}{\text{Expression 2}}\) is reframed as multiplication \(\frac{\text{Expression 1}}{1} \times \frac{1}{\text{Expression 2}}\). This involves taking the second expression, flipping the numerator and the denominator, and multiplying it with the first.
This step is crucial because it transforms a potentially complex division problem into a simpler multiplication problem, allowing you to apply your factoring and simplifying skills effectively.

Multiplication of Rational Expressions

Multiplying rational expressions is about multiplying the numerators together and the denominators together, just like with regular fractions. However, attention must be paid to factor and simplify both before and after the multiplication step.
For instance, after rewriting our division problem into a multiplication one, we multiply: \(\frac{-2(b-1)(b+2)}{4(2b+1)(b-4)} \times \frac{(b-3)(2b+1)}{(b-1)^2}\). Post simplification, factors are canceled, resulting in a simpler form \(\frac{-2(b+2)(b-3)}{4(b-4)(b-1)}\).
Efficiency in multiplication depends on meticulous attention to common factors in both the numerator and denominator, allowing for subsequent ease in simplification. Again, remember to always simplify your final answer to ensure it's as straightforward as possible!