Simplifying Rational Expressions What Is The Product

To determine the product of the given expression, we need to simplify the expression:

$$\frac{2 a-7}{a} \cdot \frac{3 a^2}{2 a^2-11 a+14}$$

First, let's factor the quadratic expression in the denominator of the second fraction:

$2 a^2 - 11 a + 14$

We are looking for two numbers that multiply to $2 * 14 = 28$ and add up to $-11$. These numbers are $-4$ and $-7$. So we can rewrite the middle term as:

$2 a^2 - 4 a - 7 a + 14$

Now, factor by grouping:

$2 a (a - 2) - 7 (a - 2)$

$(2 a - 7)(a - 2)$

Now, substitute this back into the original expression:

$$\frac{2 a-7}{a} \cdot \frac{3 a^2}{(2 a-7)(a-2)}$$

We can now cancel out the common factors:

$$\frac{(2 a-7)}{a} \cdot \frac{3 a^2}{(2 a-7)(a-2)}$$

Cancel out $(2a - 7)$ from the numerator and the denominator:

$$\frac{1}{a} \cdot \frac{3 a^2}{(a-2)}$$

Now, cancel out $a$ from the numerator and the denominator:

$$\frac{1}{1} \cdot \frac{3 a}{(a-2)}$$

So, the simplified expression is:

$$\frac{3 a}{a-2}$$

Therefore, the correct answer is:

B. $\frac{3 a}{a-2}$

Detailed Explanation of Simplifying Rational Expressions

When simplifying rational expressions, it's essential to break down each component step by step to ensure accuracy and understanding. Simplifying rational expressions involves factoring, identifying common factors, and canceling them out to arrive at the simplest form. This process is akin to reducing fractions in basic arithmetic, but with the added complexity of algebraic expressions. Let’s delve into a comprehensive explanation of each step, focusing on the given expression:

$$\frac{2 a-7}{a} \cdot \frac{3 a^2}{2 a^2-11 a+14}$$

1. Factoring the Polynomials

The first crucial step in simplifying rational expressions is to factor each polynomial in both the numerator and the denominator. Factoring involves breaking down each polynomial into its constituent factors, which are simpler expressions that, when multiplied together, yield the original polynomial. This step is vital because it reveals common factors between the numerator and the denominator that can be canceled out.

In our expression, we have:

• Numerator of the first fraction: $(2a - 7)$, which is already in its simplest form and cannot be factored further.
• Denominator of the first fraction: $a$, which is a monomial and cannot be factored further.
• Numerator of the second fraction: $3a^2$, which can be written as $3 * a * a$.
• Denominator of the second fraction: $2a^2 - 11a + 14$, which is a quadratic expression and requires factoring.

To factor the quadratic expression $2a^2 - 11a + 14$, we look for two numbers that multiply to the product of the leading coefficient (2) and the constant term (14), which is $2 * 14 = 28$, and add up to the middle coefficient (-11). These numbers are -4 and -7. Thus, we rewrite the middle term -11a as $-4a - 7a$:

$2a^2 - 11a + 14 = 2a^2 - 4a - 7a + 14$

Next, we factor by grouping:

$2a^2 - 4a - 7a + 14 = 2a(a - 2) - 7(a - 2)$

We can see that $(a - 2)$ is a common factor, so we factor it out:

$2a(a - 2) - 7(a - 2) = (2a - 7)(a - 2)$

Now, the factored form of the denominator is $(2a - 7)(a - 2)$.

2. Rewriting the Expression with Factored Forms

After factoring each polynomial, we rewrite the original expression using the factored forms:

$$\frac{2 a-7}{a} \cdot \frac{3 a^2}{2 a^2-11 a+14} = \frac{(2 a-7)}{a} \cdot \frac{3 a^2}{(2 a-7)(a-2)}$$

This step makes it easier to identify common factors between the numerators and denominators.

3. Identifying and Canceling Common Factors

The identification and cancellation of common factors is a crucial step in simplifying rational expressions. A common factor is an expression that appears in both the numerator and the denominator. Canceling these common factors simplifies the expression by reducing it to its lowest terms. In our rewritten expression:

$$\frac{(2 a-7)}{a} \cdot \frac{3 a^2}{(2 a-7)(a-2)}$$

We observe that $(2a - 7)$ is a common factor in the numerator and the denominator. We can cancel this common factor:

$$\frac{\cancel{(2 a-7)}}{a} \cdot \frac{3 a^2}{\cancel{(2 a-7)}(a-2)} = \frac{1}{a} \cdot \frac{3 a^2}{(a-2)}$$

Next, we notice that $a$ is also a common factor. In the numerator, we have $3a^2$, which can be written as $3 * a * a$. In the denominator, we have $a$. We can cancel one factor of $a$ from both the numerator and the denominator:

$$\frac{1}{\cancel{a}} \cdot \frac{3 a^{\cancel{2}}}{(a-2)} = \frac{1}{1} \cdot \frac{3 a}{(a-2)}$$

4. Simplifying the Remaining Expression

After canceling all common factors, we simplify the remaining expression by multiplying the terms in the numerator and the denominator. In our case, we have:

$$\frac{1}{1} \cdot \frac{3 a}{(a-2)} = \frac{3 a}{a-2}$$

This simplified form represents the original expression in its lowest terms. There are no more common factors to cancel, and the expression is now in its most concise form.

5. Final Answer

Therefore, the simplified expression is:

$$\frac{3 a}{a-2}$$

This matches option B from the given choices, so the correct answer is:

B. $\frac{3 a}{a-2}$

Common Mistakes to Avoid

Simplifying rational expressions can be tricky, and there are several common mistakes that students often make. Awareness of these pitfalls can help prevent errors and improve accuracy.

1. Canceling Terms Instead of Factors

One of the most common mistakes is canceling terms instead of factors. Factors are expressions that are multiplied together, while terms are expressions that are added or subtracted. You can only cancel common factors, not terms. For example, in the expression $\frac{a + 2}{a}$, you cannot cancel the $a$’s because $a$ is a term, not a factor.

2. Not Factoring Completely

Failing to factor completely can lead to missing common factors and an incorrectly simplified expression. Always ensure that all polynomials are factored to their simplest forms. For instance, if you have $2a^2 - 4a$, you should factor out $2a$ to get $2a(a - 2)$.

3. Incorrectly Factoring Quadratic Expressions

Factoring quadratic expressions can be challenging, and errors in this step can propagate through the entire simplification process. It’s important to double-check your factoring by multiplying the factors back together to ensure they match the original quadratic expression.

4. Forgetting to Distribute Negative Signs

When factoring out a negative sign or dealing with expressions involving subtraction, forgetting to distribute the negative sign is a common mistake. For example, when factoring $-1$ from $-a + b$, the correct result is $-(a - b)$, not $-a - b$.

5. Simplifying Before Factoring

Attempting to simplify before factoring can make the process more complicated and increase the likelihood of errors. Factoring first allows you to identify and cancel common factors more easily.

6. Missing Common Factors

Overlooking common factors can prevent you from simplifying the expression completely. Always carefully examine the numerators and denominators for any shared factors.

7. Errors in Arithmetic

Simple arithmetic errors, such as incorrect multiplication or addition, can lead to incorrect factoring and simplification. Double-check your calculations, especially when dealing with larger numbers or negative signs.

8. Not Rewriting the Expression After Canceling

After canceling common factors, not rewriting the expression can lead to confusion and errors in the subsequent steps. Rewriting the expression makes it clear what terms are remaining and helps prevent mistakes.

By being mindful of these common mistakes and practicing regularly, you can improve your skills in simplifying rational expressions and achieve greater accuracy.

Practice Problems

To solidify your understanding of simplifying rational expressions, working through practice problems is essential. Here are a few additional examples to help you hone your skills:

Problem 1

Simplify the expression:

$$\frac{x^2 - 4}{x^2 + 4x + 4}$$

Solution:

1. Factor the numerator and the denominator:
   • Numerator: $x^2 - 4$ is a difference of squares, so it factors as $(x - 2)(x + 2)$.
   • Denominator: $x^2 + 4x + 4$ is a perfect square trinomial, so it factors as $(x + 2)(x + 2)$.
2. Rewrite the expression with the factored forms:

   $$\frac{(x - 2)(x + 2)}{(x + 2)(x + 2)}$$

3. Cancel the common factor $(x + 2)$:

   $$\frac{(x - 2)\cancel{(x + 2)}}{(x + 2)\cancel{(x + 2)}} = \frac{x - 2}{x + 2}$$

So, the simplified expression is $\frac{x - 2}{x + 2}$.

Problem 2

Simplify the expression:

$$\frac{2y^2 + 6y}{y^2 + 5y + 6}$$

Solution:

1. Factor the numerator and the denominator:
   • Numerator: $2y^2 + 6y$ can be factored by taking out the common factor $2y$, resulting in $2y(y + 3)$.
   • Denominator: $y^2 + 5y + 6$ factors as $(y + 2)(y + 3)$.
2. Rewrite the expression with the factored forms:

   $$\frac{2y(y + 3)}{(y + 2)(y + 3)}$$

3. Cancel the common factor $(y + 3)$:

   $$\frac{2y\cancel{(y + 3)}}{(y + 2)\cancel{(y + 3)}} = \frac{2y}{y + 2}$$

Thus, the simplified expression is $\frac{2y}{y + 2}$.

Problem 3

Simplify the expression:

$$\frac{3a^2 - 12}{a^2 - 4a + 4}$$

Solution:

1. Factor the numerator and the denominator:
   • Numerator: $3a^2 - 12$ can be factored by first taking out the common factor $3$, resulting in $3(a^2 - 4)$. Then, $a^2 - 4$ is a difference of squares, so it factors as $(a - 2)(a + 2)$. Thus, the numerator becomes $3(a - 2)(a + 2)$.
   • Denominator: $a^2 - 4a + 4$ is a perfect square trinomial, so it factors as $(a - 2)(a - 2)$.
2. Rewrite the expression with the factored forms:

   $$\frac{3(a - 2)(a + 2)}{(a - 2)(a - 2)}$$

3. Cancel the common factor $(a - 2)$:

   $$\frac{3\cancel{(a - 2)}(a + 2)}{(a - 2)\cancel{(a - 2)}} = \frac{3(a + 2)}{a - 2}$$

Therefore, the simplified expression is $\frac{3(a + 2)}{a - 2}$.

By working through these practice problems and reviewing the steps, you can gain confidence in simplifying rational expressions. Remember to always factor first, identify common factors, and cancel them to reach the simplest form. Practice makes perfect, so keep at it! Simplify complex algebraic fractions effectively by identifying and canceling common factors through factoring. This makes problems more manageable and easier to solve.