A) Factorize $x^2 - 2x - 3$.b) Hence, Simplify 2 X − 1 X 2 − 2 X − 3 + 1 X − 3 \frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3} X 2 − 2 X − 3 2 X − 1 ​ + X − 3 1 ​ .

Introduction

In mathematics, factorizing quadratic expressions and simplifying rational functions are essential skills that are used extensively in various fields, including algebra, calculus, and engineering. In this article, we will focus on factorizing the quadratic expression $x^2 - 2x - 3$ and then use this factorization to simplify the rational function $\frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3}$.

Factorizing Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has the highest power of the variable as two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. To factorize a quadratic expression, we need to find two binomials whose product is equal to the quadratic expression.

Factorizing $x^2 - 2x - 3$

To factorize the quadratic expression $x^2 - 2x - 3$, we need to find two binomials whose product is equal to this expression. We can start by looking for two numbers whose product is $-3$ and whose sum is $-2$. These numbers are $-3$ and $1$, because $(-3) \times (1) = -3$ and $(-3) + (1) = -2$.

Using these numbers, we can write the quadratic expression as:

$$x^2 - 2x - 3 = (x - 3)(x + 1)$$

Therefore, the factorization of the quadratic expression $x^2 - 2x - 3$ is $(x - 3)(x + 1)$.

Simplifying Rational Functions

A rational function is a function that is expressed as the ratio of two polynomials. To simplify a rational function, we need to find a common denominator and then combine the fractions.

Simplifying $\frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3}$

To simplify the rational function $\frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3}$, we need to find a common denominator. The common denominator is the product of the two denominators, which is $(x - 3)(x + 1)$.

We can rewrite the rational function as:

$$\frac{2x - 1}{(x - 3)(x + 1)} + \frac{1}{(x - 3)(x + 1)}$$

Now, we can combine the fractions by adding the numerators:

$$\frac{(2x - 1) + 1}{(x - 3)(x + 1)}$$

Simplifying the numerator, we get:

$$\frac{2x - 1 + 1}{(x - 3)(x + 1)} = \frac{2x}{(x - 3)(x + 1)}$$

Therefore, the simplified form of the rational function $\frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3}$ is $\frac{2x}{(x - 3)(x + 1)}$.

Conclusion

In this article, we factorized the quadratic expression $x^2 - 2x - 3$ and then used this factorization to simplify the rational function $\frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3}$. We found that the factorization of the quadratic expression is $(x - 3)(x + 1)$ and the simplified form of the rational function is $\frac{2x}{(x - 3)(x + 1)}$.

Final Answer

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Introduction

In our previous article, we factorized the quadratic expression $x^2 - 2x - 3$ and then used this factorization to simplify the rational function $\frac{2x - 1}{x^2 - 2x - 3} + \frac{1}{x - 3}$. In this article, we will answer some frequently asked questions related to factorizing quadratic expressions and simplifying rational functions.

Q&A

Q: What is the difference between factorizing and simplifying?

A: Factorizing is the process of expressing a quadratic expression as a product of two binomials, while simplifying is the process of combining fractions with a common denominator.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two binomials whose product is equal to the quadratic expression. You can start by looking for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the common denominator in simplifying rational functions?

A: The common denominator is the product of the two denominators. For example, if you have two fractions with denominators $x - 3$ and $x + 1$, the common denominator is $(x - 3)(x + 1)$.

Q: How do I simplify a rational function with a common denominator?

A: To simplify a rational function with a common denominator, you need to combine the fractions by adding the numerators. You can then simplify the resulting fraction by canceling out any common factors.

Q: What are some common mistakes to avoid when factorizing quadratic expressions?

A: Some common mistakes to avoid when factorizing quadratic expressions include:

• Not checking if the quadratic expression can be factored
• Not using the correct method to factorize the quadratic expression
• Not simplifying the resulting expression

Q: What are some common mistakes to avoid when simplifying rational functions?

A: Some common mistakes to avoid when simplifying rational functions include:

• Not finding a common denominator
• Not combining the fractions correctly
• Not simplifying the resulting fraction

Tips and Tricks

Tip 1: Use the correct method to factorize quadratic expressions

When factorizing quadratic expressions, make sure to use the correct method. For example, if the quadratic expression can be factored using the difference of squares formula, use that formula.

Tip 2: Simplify the resulting fraction

When simplifying rational functions, make sure to simplify the resulting fraction. This can be done by canceling out any common factors.

Tip 3: Check your work

When factorizing quadratic expressions or simplifying rational functions, make sure to check your work. This can be done by plugging in values or using a calculator.

Conclusion

In this article, we answered some frequently asked questions related to factorizing quadratic expressions and simplifying rational functions. We also provided some tips and tricks to help you avoid common mistakes and simplify rational functions correctly.

Final Answer

The final answer is $\boxed{\frac{2x}{(x - 3)(x + 1)}}$.