Simplify The Expression:${ \frac{6x^2 + 17x}{(x+3)(x+2)} + \frac{x^2 + X}{x^2 + 8x + 15} + \frac{x}{x+3} }$

Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves breaking down intricate expressions into simpler forms, making them easier to work with and understand. In this article, we will focus on simplifying the given expression: ${ \frac{6x^2 + 17x}{(x+3)(x+2)} + \frac{x^2 + x}{x^2 + 8x + 15} + \frac{x}{x+3} }$. We will break down the expression into manageable parts, apply various algebraic techniques, and simplify it step by step.

Breaking Down the Expression

The given expression consists of three fractions, each with a different denominator. To simplify the expression, we need to find a common denominator for all three fractions. The denominators are $(x+3)(x+2)$, $x^2 + 8x + 15$, and $x+3$. We can start by factoring the second denominator:

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

Now, we have the denominators in factored form: $(x+3)(x+2)$, $(x+3)(x+5)$, and $x+3$.

Finding a Common Denominator

To find a common denominator, we need to multiply all the denominators together:

$(x+3)(x+2) \times (x+3)(x+5) \times (x+3) = (x+3)^3(x+2)(x+5)$

This is the common denominator for all three fractions.

Simplifying Each Fraction

Now that we have a common denominator, we can simplify each fraction individually. We will start with the first fraction:

$\frac{6x^2 + 17x}{(x+3)(x+2)}$

We can factor the numerator:

$6x^2 + 17x = x(6x+17)$

Now, we can rewrite the fraction:

$\frac{x(6x+17)}{(x+3)(x+2)}$

Simplifying the Second Fraction

The second fraction is:

$\frac{x^2 + x}{x^2 + 8x + 15}$

We can factor the numerator:

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

And the denominator:

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

Now, we can rewrite the fraction:

$\frac{x(x+1)}{(x+3)(x+5)}$

Simplifying the Third Fraction

The third fraction is:

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

This fraction is already simplified.

Combining the Fractions

Now that we have simplified each fraction, we can combine them using the common denominator:

$\frac{x(6x+17)}{(x+3)^2(x+2)} + \frac{x(x+1)}{(x+3)^2(x+5)} + \frac{x}{(x+3)^2}$

We can combine the fractions by finding a common denominator for the numerators:

$\frac{x(6x+17)(x+5) + x(x+1)(x+2) + x(x+3)^2}{(x+3)^2(x+2)(x+5)}$

Simplifying the Numerator

Now, we can simplify the numerator:

$x(6x+17)(x+5) + x(x+1)(x+2) + x(x+3)^2$

We can expand and simplify the expression:

$6x^3 + 17x^2 + 30x^2 + 85x + x^3 + x^2 + 2x + x^3 + 6x^2 + 9x$

Combine like terms:

$6x^3 + x^3 + x^3 + 30x^2 + 17x^2 + x^2 + 6x^2 + 85x + 2x + 9x$

Simplify further:

$8x^3 + 54x^2 + 96x$

Final Simplification

Now, we can rewrite the expression with the simplified numerator:

$\frac{8x^3 + 54x^2 + 96x}{(x+3)^2(x+2)(x+5)}$

This is the final simplified expression.

Conclusion

Simplifying complex algebraic expressions requires patience, persistence, and a thorough understanding of algebraic techniques. By breaking down the expression into manageable parts, applying various algebraic techniques, and simplifying step by step, we can simplify even the most intricate expressions. In this article, we simplified the given expression using factoring, finding a common denominator, and combining fractions. The final simplified expression is $\frac{8x^3 + 54x^2 + 96x}{(x+3)^2(x+2)(x+5)}$.

Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. In our previous article, we walked through the step-by-step process of simplifying the given expression: ${ \frac{6x^2 + 17x}{(x+3)(x+2)} + \frac{x^2 + x}{x^2 + 8x + 15} + \frac{x}{x+3} }$. In this article, we will address some common questions and concerns that students may have when simplifying complex algebraic expressions.

Q&A

Q: What is the first step in simplifying a complex algebraic expression?

A: The first step in simplifying a complex algebraic expression is to break it down into manageable parts. This involves identifying the different components of the expression, such as the numerator and denominator, and factoring them if possible.

Q: How do I find a common denominator for multiple fractions?

A: To find a common denominator for multiple fractions, you need to multiply all the denominators together. This will give you a common denominator that all the fractions can share.

Q: What is the difference between factoring and simplifying an expression?

A: Factoring an expression involves breaking it down into its simplest form by identifying the common factors. Simplifying an expression, on the other hand, involves reducing it to its simplest form by combining like terms and eliminating any unnecessary components.

Q: How do I know when to use the distributive property when simplifying an expression?

A: The distributive property is used when you need to multiply a single term by multiple terms. For example, if you have the expression $2(x+3)$, you would use the distributive property to multiply the 2 by each term inside the parentheses.

Q: Can I simplify an expression by canceling out terms?

A: Yes, you can simplify an expression by canceling out terms. However, you need to make sure that the terms you are canceling out are actually present in the expression and that the cancellation is valid.

Q: How do I know when to use the commutative property when simplifying an expression?

A: The commutative property is used when you need to rearrange the terms in an expression. For example, if you have the expression $x+3$, you can use the commutative property to rearrange the terms to get $3+x$.

Q: Can I simplify an expression by combining like terms?

A: Yes, you can simplify an expression by combining like terms. This involves adding or subtracting terms that have the same variable and exponent.

Q: How do I know when to use the associative property when simplifying an expression?

A: The associative property is used when you need to group terms in an expression. For example, if you have the expression $(x+2)+(x+3)$, you can use the associative property to group the terms as $(x+x)+(2+3)$.

Q: Can I simplify an expression by canceling out a common factor?

A: Yes, you can simplify an expression by canceling out a common factor. However, you need to make sure that the common factor is actually present in the expression and that the cancellation is valid.

Conclusion

Simplifying complex algebraic expressions requires patience, persistence, and a thorough understanding of algebraic techniques. By addressing common questions and concerns, we hope to have provided a better understanding of the simplification process. Remember to break down the expression into manageable parts, find a common denominator, and combine like terms to simplify the expression.