Restrictions for the Variable

Oct 13, 2025 by miodeya.com 0 views

Introduction

In this article, we will explore the process of multiplying two algebraic expressions, specifically the given expression: $\frac{3x^2+9x+6}{2x-6} \cdot \frac{5x^2+5x-60}{6x^2+18x+12}$ . We will also identify the restrictions for the variable and simplify the resulting expression.

Step 1: Factor the Numerators and Denominators

To multiply the given expressions, we first need to factor the numerators and denominators. The factored forms will help us simplify the expression and identify any common factors.

Factoring the Numerators

The first numerator, $3x^2+9x+6$ , can be factored as follows:

3x^2+9x+6 = 3(x^2+3x+2) = 3(x+1)(x+2)

The second numerator, $5x^2+5x-60$ , can be factored as follows:

5x^2+5x-60 = 5(x^2+x-12) = 5(x+4)(x-3)

Factoring the Denominators

The first denominator, $2x-6$ , can be factored as follows:

2x-6 = 2(x-3)

The second denominator, $6x^2+18x+12$ , can be factored as follows:

6x^2+18x+12 = 6(x^2+3x+2) = 6(x+1)(x+2)

Step 2: Multiply the Factored Expressions

Now that we have factored the numerators and denominators, we can multiply the expressions.

\frac{3(x+1)(x+2)}{2(x-3)} \cdot \frac{5(x+4)(x-3)}{6(x+1)(x+2)}

We can cancel out the common factors in the numerator and denominator.

\frac{3(x+1)(x+2)}{2(x-3)} \cdot \frac{5(x+4)(x-3)}{6(x+1)(x+2)} = \frac{3 \cdot 5 \cdot (x+4)}{2 \cdot 6}

Step 3: Simplify the Expression

We can simplify the expression by canceling out any common factors.

\frac{3 \cdot 5 \cdot (x+4)}{2 \cdot 6} = \frac{15(x+4)}{12}

We can further simplify the expression by dividing both the numerator and denominator by their greatest common factor, which is 3.

\frac{15(x+4)}{12} = \frac{5(x+4)}{4}

Restrictions for the Variable

The variable $x$ has restrictions in the original expression. We need to identify these restrictions and list them.

The first expression has a denominator of $2x-6$ , which means that $x$ cannot be equal to $\frac{6}{2} = 3$ . This is because division by zero is undefined.

The second expression has a denominator of $6x^2+18x+12$ , which can be factored as $6(x+1)(x+2)$ . This means that $x$ cannot be equal to $-1$ or $-2$ , as these values would make the denominator zero.

Therefore, the restrictions for the variable $x$ are:

$x \neq 3$
$x \neq -1$
$x \neq -2$

Conclusion

Introduction

In our previous article, we explored the process of multiplying two algebraic expressions, specifically the given expression: $\frac{3x^2+9x+6}{2x-6} \cdot \frac{5x^2+5x-60}{6x^2+18x+12}$ . We also identified the restrictions for the variable and simplified the resulting expression. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the purpose of factoring the numerators and denominators?

A: Factoring the numerators and denominators helps us simplify the expression and identify any common factors. This can make it easier to cancel out common factors and simplify the expression.

Q: Why do we need to identify the restrictions for the variable?

A: We need to identify the restrictions for the variable because they affect the validity of the expression. If the variable is not within the restrictions, the expression may be undefined or invalid.

Q: Can we multiply the expressions without factoring the numerators and denominators?

A: Yes, we can multiply the expressions without factoring the numerators and denominators. However, factoring can make it easier to identify common factors and simplify the expression.

Q: What is the difference between a restriction and a solution?

A: A restriction is a value that the variable cannot take, whereas a solution is a value that the variable can take. In the context of the given expression, the restrictions are $x \neq 3$ , $x \neq -1$ , and $x \neq -2$ , whereas the solutions are all real numbers except for these values.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by canceling out any common factors. However, in this case, we have already simplified the expression as much as possible.

Q: How do we know when to stop simplifying the expression?

A: We know when to stop simplifying the expression when we have canceled out all common factors and the expression cannot be simplified further.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{5(x+4)}{4}$ .

Q: What are the restrictions for the variable?

A: The restrictions for the variable are $x \neq 3$ , $x \neq -1$ , and $x \neq -2$ .

Conclusion

In this article, we answered some frequently asked questions related to multiplying the given expression and identifying the restrictions for the variable. We also reviewed the process of factoring the numerators and denominators, canceling out common factors, and simplifying the expression. We hope that this article has been helpful in clarifying any doubts or questions you may have had.

Additional Resources

If you would like to learn more about multiplying algebraic expressions and identifying restrictions, we recommend the following resources:

Khan Academy: Multiplying Algebraic Expressions
Mathway: Multiplying Algebraic Expressions
Wolfram Alpha: Multiplying Algebraic Expressions

We hope that these resources are helpful in your learning journey.