Simplify The Expression:$\[ \frac{12-x-x^2}{x^2-8x+15} \\]

Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of rational expressions and explore the steps involved in simplifying the given expression: $\frac{12-x-x^2}{x^2-8x+15}$. We will break down the problem into manageable parts, using factoring and simplification techniques to arrive at the final answer.

Understanding Rational Expressions

Before we dive into the problem, let's take a moment to understand what rational expressions are. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression.

Factoring the Numerator

The first step in simplifying the given expression is to factor the numerator. The numerator is $12-x-x^2$, which can be factored as follows:

$$12-x-x^2 = -(x^2+x-12)$$

We can factor the quadratic expression $x^2+x-12$ by finding two numbers whose product is $-12$ and whose sum is $1$. The numbers are $-4$ and $3$, so we can write:

$$x^2+x-12 = (x-4)(x+3)$$

Therefore, the factored form of the numerator is:

$$12-x-x^2 = -(x-4)(x+3)$$

Factoring the Denominator

The next step is to factor the denominator. The denominator is $x^2-8x+15$, which can be factored as follows:

$$x^2-8x+15 = (x-5)(x-3)$$

Simplifying the Expression

Now that we have factored the numerator and denominator, we can simplify the expression by canceling out common factors. The expression is:

$$\frac{12-x-x^2}{x^2-8x+15} = \frac{-(x-4)(x+3)}{(x-5)(x-3)}$$

We can cancel out the common factor $(x-3)$ from the numerator and denominator, leaving us with:

$$\frac{-(x-4)(x+3)}{(x-5)(x-3)} = \frac{-(x-4)(x+3)}{x-5}$$

Final Answer

The final answer is $\frac{-(x-4)(x+3)}{x-5}$. However, we can simplify this expression further by factoring the numerator and canceling out common factors.

Simplifying the Final Answer

The numerator is $-(x-4)(x+3)$, which can be expanded as follows:

$$-(x-4)(x+3) = -x^2-x+12$$

We can factor the quadratic expression $-x^2-x+12$ by finding two numbers whose product is $12$ and whose sum is $-1$. The numbers are $-4$ and $3$, so we can write:

$$-x^2-x+12 = -(x^2+x-12) = -(x-4)(x+3)$$

Therefore, the factored form of the numerator is:

$$-x^2-x+12 = -(x-4)(x+3)$$

We can cancel out the common factor $(x-3)$ from the numerator and denominator, leaving us with:

$$\frac{-(x-4)(x+3)}{x-5} = \frac{-(x-4)(x+3)}{x-5}$$

However, we can simplify this expression further by factoring the numerator and canceling out common factors.

Final Answer

The final answer is $\frac{-(x-4)(x+3)}{x-5}$. However, we can simplify this expression further by factoring the numerator and canceling out common factors.

Conclusion

In this article, we have simplified the given expression $\frac{12-x-x^2}{x^2-8x+15}$ by factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression. We have shown that the final answer is $\frac{-(x-4)(x+3)}{x-5}$, which can be simplified further by factoring the numerator and canceling out common factors.

Frequently Asked Questions

• What is a rational expression? A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.
• How do I simplify a rational expression? To simplify a rational expression, you need to factor the numerator and denominator, cancel out common factors, and simplify the resulting expression.
• What is the final answer to the given expression? The final answer to the given expression is $\frac{-(x-4)(x+3)}{x-5}$.

References

• [1] Algebra, 2nd edition, by Michael Artin
• [2] Calculus, 3rd edition, by Michael Spivak
• [3] Rational Expressions, by Math Open Reference

Related Articles

• Simplifying Rational Expressions: A Step-by-Step Guide
• Factoring Quadratic Expressions: A Step-by-Step Guide
• Simplifying Algebraic Expressions: A Step-by-Step Guide
  

Introduction

In our previous article, we simplified the expression $\frac{12-x-x^2}{x^2-8x+15}$ by factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression. In this article, we will answer some frequently asked questions related to simplifying rational expressions.

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out common factors, and simplify the resulting expression.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can I simplify a rational expression by canceling out common factors in the numerator and denominator?

A: Yes, you can simplify a rational expression by canceling out common factors in the numerator and denominator. However, you need to make sure that the common factors are not canceled out completely, as this would result in an undefined expression.

Q: How do I know if a rational expression is undefined?

A: A rational expression is undefined if the denominator is equal to zero. This is because division by zero is undefined in mathematics.

Q: Can I simplify a rational expression by combining like terms in the numerator and denominator?

A: Yes, you can simplify a rational expression by combining like terms in the numerator and denominator. However, you need to make sure that the like terms are combined correctly and that the resulting expression is simplified.

Q: How do I know if a rational expression is in its simplest form?

A: A rational expression is in its simplest form if it cannot be simplified further by canceling out common factors or combining like terms.

Q: Can I simplify a rational expression by using algebraic identities?

A: Yes, you can simplify a rational expression by using algebraic identities. For example, you can use the identity $(a+b)^2 = a^2 + 2ab + b^2$ to simplify a rational expression.

Q: How do I know if a rational expression is a polynomial?

A: A rational expression is a polynomial if it can be expressed as a sum of terms, where each term is a product of a variable and a constant.

Q: Can I simplify a rational expression by using the distributive property?

A: Yes, you can simplify a rational expression by using the distributive property. For example, you can use the distributive property to simplify a rational expression by multiplying each term in the numerator by each term in the denominator.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying rational expressions. We have shown that simplifying rational expressions is an important skill in algebra and that it can be used to solve a wide range of problems.

Frequently Asked Questions

• What is a rational expression?
• How do I simplify a rational expression?
• What is the difference between a rational expression and a rational number?
• Can I simplify a rational expression by canceling out common factors in the numerator and denominator?
• How do I know if a rational expression is undefined?
• Can I simplify a rational expression by combining like terms in the numerator and denominator?
• How do I know if a rational expression is in its simplest form?
• Can I simplify a rational expression by using algebraic identities?
• How do I know if a rational expression is a polynomial?
• Can I simplify a rational expression by using the distributive property?

References

• [1] Algebra, 2nd edition, by Michael Artin
• [2] Calculus, 3rd edition, by Michael Spivak
• [3] Rational Expressions, by Math Open Reference

Related Articles

• Simplifying Rational Expressions: A Step-by-Step Guide
• Factoring Quadratic Expressions: A Step-by-Step Guide
• Simplifying Algebraic Expressions: A Step-by-Step Guide