Evaluate The Following Expression:${ \frac{2x^3 + 5x^2 - 3x + 4}{x+2} }$

Mar 9, 2025 by ADMIN 74 views

**Evaluating the Given Algebraic Expression**

Introduction

In this article, we will delve into the process of evaluating a given algebraic expression, which involves simplifying the expression by performing various mathematical operations. The expression we will be working with is $\frac{2x^3 + 5x^2 - 3x + 4}{x+2}$ . Our goal is to simplify this expression and understand its behavior.

Understanding the Expression

The given expression is a rational expression, which means it is the ratio of two polynomials. The numerator of the expression is $2x^3 + 5x^2 - 3x + 4$ , and the denominator is $x+2$ . To evaluate this expression, we need to simplify it by performing various mathematical operations.

Simplifying the Expression

To simplify the expression, we can start by factoring the numerator and denominator. Factoring the numerator, we get:

\begin{aligned} 2x^3 + 5x^2 - 3x + 4 &= (x+2)(2x^2 + 3x - 2) \\ &= (x+2)(2x-1)(x+2) \end{aligned}

Now, we can cancel out the common factor of $(x+2)$ from the numerator and denominator:

\begin{aligned} \frac{(x+2)(2x-1)(x+2)}{x+2} &= (2x-1)(x+2) \\ &= 2x^2 + 3x - 2 \end{aligned}

Analyzing the Simplified Expression

The simplified expression is $2x^2 + 3x - 2$ . This is a quadratic expression, which means it can be analyzed using various techniques such as factoring, completing the square, or using the quadratic formula.

Factoring the Quadratic Expression

We can try to factor the quadratic expression $2x^2 + 3x - 2$ :

\begin{aligned} 2x^2 + 3x - 2 &= (2x-1)(x+2) \end{aligned}

Completing the Square

We can also try to complete the square for the quadratic expression $2x^2 + 3x - 2$ :

\begin{aligned} 2x^2 + 3x - 2 &= 2(x^2 + \frac{3}{2}x) - 2 \\ &= 2(x^2 + \frac{3}{2}x + \frac{9}{16}) - 2 - \frac{9}{8} \\ &= 2(x + \frac{3}{4})^2 - \frac{25}{8} \end{aligned}

Using the Quadratic Formula

We can also use the quadratic formula to solve the quadratic expression $2x^2 + 3x - 2$ :

\begin{aligned} x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ &= \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} \\ &= \frac{-3 \pm \sqrt{9 + 16}}{4} \\ &= \frac{-3 \pm \sqrt{25}}{4} \\ &= \frac{-3 \pm 5}{4} \end{aligned}

Conclusion

In this article, we evaluated the given algebraic expression $\frac{2x^3 + 5x^2 - 3x + 4}{x+2}$ by simplifying it using various mathematical operations. We factored the numerator and denominator, canceled out common factors, and analyzed the simplified expression using techniques such as factoring, completing the square, and using the quadratic formula. Our goal was to understand the behavior of the expression and simplify it to its most basic form.

Final Answer

The final answer to the given expression is $2x^2 + 3x - 2$ . This is a quadratic expression that can be analyzed using various techniques such as factoring, completing the square, or using the quadratic formula.

Recommendations

Based on our analysis, we recommend the following:

Use factoring to simplify the expression and cancel out common factors.
Use completing the square to analyze the quadratic expression and understand its behavior.
Use the quadratic formula to solve the quadratic expression and find its roots.

By following these recommendations, you can simplify the given expression and understand its behavior.

Introduction

In our previous article, we evaluated the given algebraic expression $\frac{2x^3 + 5x^2 - 3x + 4}{x+2}$ by simplifying it using various mathematical operations. We factored the numerator and denominator, canceled out common factors, and analyzed the simplified expression using techniques such as factoring, completing the square, and using the quadratic formula. In this article, we will answer some frequently asked questions related to the given expression.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $2x^2 + 3x - 2$ . This is a quadratic expression that can be analyzed using various techniques such as factoring, completing the square, or using the quadratic formula.

Q: How do I simplify the given expression?

A: To simplify the given expression, you can start by factoring the numerator and denominator. Then, cancel out common factors and analyze the simplified expression using techniques such as factoring, completing the square, or using the quadratic formula.

Q: What is the difference between factoring and completing the square?

A: Factoring involves expressing an expression as a product of simpler expressions, while completing the square involves expressing an expression in the form $(x + a)^2 + b$ . Both techniques can be used to simplify and analyze quadratic expressions.

Q: How do I use the quadratic formula to solve the given expression?

A: To use the quadratic formula to solve the given expression, you can plug in the values of $a$ , $b$ , and $c$ into the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Then, simplify the expression and find the roots of the quadratic equation.

Q: What are the roots of the given expression?

A: The roots of the given expression are $\frac{-3 + 5}{4} = 1/2$ and $\frac{-3 - 5}{4} = -2$ . These are the values of $x$ that make the expression equal to zero.

Q: How do I analyze the behavior of the given expression?

A: To analyze the behavior of the given expression, you can use various techniques such as factoring, completing the square, or using the quadratic formula. You can also graph the expression to visualize its behavior.

Q: What are the applications of the given expression?

A: The given expression has various applications in mathematics and science. It can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits.