Simplify The Expression.${ \frac{x^2 + 8x + 12}{x^2 + 7x + 6} = \square }$Question Help:- Video- Written Example

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. In this article, we will focus on simplifying a given expression involving polynomials. We will break down the process into manageable steps and provide a clear explanation of each step.

The Given Expression

The given expression is:

$$\frac{x^2 + 8x + 12}{x^2 + 7x + 6} = \square$$

Our goal is to simplify this expression by factoring the numerator and denominator, and then canceling out any common factors.

Step 1: Factor the Numerator

To factor the numerator, we need to find two numbers whose product is 12 and whose sum is 8. These numbers are 6 and 2, since 6 × 2 = 12 and 6 + 2 = 8. Therefore, we can write the numerator as:

$$x^2 + 8x + 12 = (x + 6)(x + 2)$$

Step 2: Factor the Denominator

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. In this article, we will focus on simplifying a given expression involving polynomials. We will break down the process into manageable steps and provide a clear explanation of each step.

The Given Expression

The given expression is:

$$\frac{x^2 + 8x + 12}{x^2 + 7x + 6} = \square$$

Our goal is to simplify this expression by factoring the numerator and denominator, and then canceling out any common factors.

Step 1: Factor the Numerator

To factor the numerator, we need to find two numbers whose product is 12 and whose sum is 8. These numbers are 6 and 2, since 6 × 2 = 12 and 6 + 2 = 8. Therefore, we can write the numerator as:

$$x^2 + 8x + 12 = (x + 6)(x + 2)$$

Step 2: Factor the Denominator

To factor the denominator, we need to find two numbers whose product is 6 and whose sum is 7. These numbers are 3 and 2, since 3 × 2 = 6 and 3 + 2 = 5. Therefore, we can write the denominator as:

$$x^2 + 7x + 6 = (x + 3)(x + 2)$$

Step 3: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we can see that both the numerator and denominator have a common factor of (x + 2). Therefore, we can cancel out this common factor:

$$\frac{(x + 6)(x + 2)}{(x + 3)(x + 2)} = \frac{x + 6}{x + 3}$$

Conclusion

In this article, we have simplified the given expression by factoring the numerator and denominator, and then canceling out any common factors. The simplified expression is:

$$\frac{x + 6}{x + 3}$$

This expression is now in its simplest form, and we can use it to solve equations and inequalities.

Q&A

Q: What is the first step in simplifying an expression? A: The first step in simplifying an expression is to factor the numerator and denominator.

Q: How do I factor the numerator and denominator? A: To factor the numerator and denominator, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the x term.

Q: What is the next step after factoring the numerator and denominator? A: The next step after factoring the numerator and denominator is to cancel out any common factors.

Q: How do I cancel out common factors? A: To cancel out common factors, you need to identify the common factors in the numerator and denominator, and then divide both the numerator and denominator by these common factors.

Q: What is the final step in simplifying an expression? A: The final step in simplifying an expression is to write the expression in its simplest form.

Q: Can I use this method to simplify any expression? A: Yes, you can use this method to simplify any expression that involves polynomials.

Q: What if I have a complex expression with multiple terms? A: If you have a complex expression with multiple terms, you can use the same method to simplify each term separately, and then combine the simplified terms.

Q: Can I use this method to solve equations and inequalities? A: Yes, you can use this method to solve equations and inequalities by simplifying the expressions involved.

Q: What if I get stuck or make a mistake? A: If you get stuck or make a mistake, don't worry! You can always go back and review the steps, or ask for help from a teacher or tutor.