What Is The Difference?$\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9}$A. $\frac{(x+5)(x+2)}{x^3-9x}$B. $\frac{(x+5)(x+4)}{x^3-9x}$C. $\frac{-2x+11}{x^3-12x-9}$D. $\frac{3(x+2)}{x^2-3x}$

by ADMIN 198 views

Introduction

In mathematics, algebraic expressions are a fundamental concept that plays a crucial role in solving various mathematical problems. The given expression, 2x+5x2βˆ’3xβˆ’3x+5x3βˆ’9xβˆ’x+1x2βˆ’9\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9}, is a complex algebraic expression that requires a thorough understanding of algebraic manipulation and simplification techniques. In this article, we will delve into the world of algebra and explore the differences between the given expression and the provided options.

Understanding the Given Expression

The given expression is a combination of three fractions, each with a different denominator. To simplify this expression, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of x2βˆ’3xx^2-3x, x3βˆ’9xx^3-9x, and x2βˆ’9x^2-9 is x3βˆ’9xx^3-9x.

Simplifying the Expression

To simplify the expression, we need to rewrite each fraction with the common denominator x3βˆ’9xx^3-9x. We can do this by multiplying the numerator and denominator of each fraction by the necessary factors.

2x+5x2βˆ’3x=(2x+5)(x)(x2βˆ’3x)(x)=(2x+5)(x)x3βˆ’3x2\frac{2x+5}{x^2-3x} = \frac{(2x+5)(x)}{(x^2-3x)(x)} = \frac{(2x+5)(x)}{x^3-3x^2}

3x+5x3βˆ’9x=(3x+5)(x2)(x3βˆ’9x)(x2)=(3x+5)(x2)x5βˆ’9x3\frac{3x+5}{x^3-9x} = \frac{(3x+5)(x^2)}{(x^3-9x)(x^2)} = \frac{(3x+5)(x^2)}{x^5-9x^3}

x+1x2βˆ’9=(x+1)(x)(x2βˆ’9)(x)=(x+1)(x)x3βˆ’9x\frac{x+1}{x^2-9} = \frac{(x+1)(x)}{(x^2-9)(x)} = \frac{(x+1)(x)}{x^3-9x}

Combining the Fractions

Now that we have rewritten each fraction with the common denominator x3βˆ’9xx^3-9x, we can combine them by adding and subtracting the numerators.

(2x+5)(x)x3βˆ’3x2βˆ’(3x+5)(x2)x5βˆ’9x3βˆ’(x+1)(x)x3βˆ’9x\frac{(2x+5)(x)}{x^3-3x^2} - \frac{(3x+5)(x^2)}{x^5-9x^3} - \frac{(x+1)(x)}{x^3-9x}

Simplifying the Numerators

To simplify the numerators, we need to multiply the terms in each numerator.

(2x+5)(x)x3βˆ’3x2=2x2+5xx3βˆ’3x2\frac{(2x+5)(x)}{x^3-3x^2} = \frac{2x^2+5x}{x^3-3x^2}

(3x+5)(x2)x5βˆ’9x3=3x3+5x2x5βˆ’9x3\frac{(3x+5)(x^2)}{x^5-9x^3} = \frac{3x^3+5x^2}{x^5-9x^3}

(x+1)(x)x3βˆ’9x=x2+xx3βˆ’9x\frac{(x+1)(x)}{x^3-9x} = \frac{x^2+x}{x^3-9x}

Combining the Numerators

Now that we have simplified the numerators, we can combine them by adding and subtracting the terms.

2x2+5xx3βˆ’3x2βˆ’3x3+5x2x5βˆ’9x3βˆ’x2+xx3βˆ’9x\frac{2x^2+5x}{x^3-3x^2} - \frac{3x^3+5x^2}{x^5-9x^3} - \frac{x^2+x}{x^3-9x}

Finding a Common Denominator

To combine the fractions, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of x3βˆ’3x2x^3-3x^2, x5βˆ’9x3x^5-9x^3, and x3βˆ’9xx^3-9x is x5βˆ’9x3x^5-9x^3.

Rewriting the Fractions

To rewrite the fractions with the common denominator x5βˆ’9x3x^5-9x^3, we need to multiply the numerator and denominator of each fraction by the necessary factors.

2x2+5xx3βˆ’3x2=(2x2+5x)(x2)(x3βˆ’3x2)(x2)=(2x2+5x)(x2)x5βˆ’3x4\frac{2x^2+5x}{x^3-3x^2} = \frac{(2x^2+5x)(x^2)}{(x^3-3x^2)(x^2)} = \frac{(2x^2+5x)(x^2)}{x^5-3x^4}

3x3+5x2x5βˆ’9x3=(3x3+5x2)(x2)(x5βˆ’9x3)(x2)=(3x3+5x2)(x2)x7βˆ’9x5\frac{3x^3+5x^2}{x^5-9x^3} = \frac{(3x^3+5x^2)(x^2)}{(x^5-9x^3)(x^2)} = \frac{(3x^3+5x^2)(x^2)}{x^7-9x^5}

x2+xx3βˆ’9x=(x2+x)(x2)(x3βˆ’9x)(x2)=(x2+x)(x2)x5βˆ’9x3\frac{x^2+x}{x^3-9x} = \frac{(x^2+x)(x^2)}{(x^3-9x)(x^2)} = \frac{(x^2+x)(x^2)}{x^5-9x^3}

Combining the Fractions

Now that we have rewritten each fraction with the common denominator x5βˆ’9x3x^5-9x^3, we can combine them by adding and subtracting the numerators.

(2x2+5x)(x2)x5βˆ’3x4βˆ’(3x3+5x2)(x2)x7βˆ’9x5βˆ’(x2+x)(x2)x5βˆ’9x3\frac{(2x^2+5x)(x^2)}{x^5-3x^4} - \frac{(3x^3+5x^2)(x^2)}{x^7-9x^5} - \frac{(x^2+x)(x^2)}{x^5-9x^3}

Simplifying the Numerators

To simplify the numerators, we need to multiply the terms in each numerator.

(2x2+5x)(x2)x5βˆ’3x4=2x4+5x3x5βˆ’3x4\frac{(2x^2+5x)(x^2)}{x^5-3x^4} = \frac{2x^4+5x^3}{x^5-3x^4}

(3x3+5x2)(x2)x7βˆ’9x5=3x5+5x4x7βˆ’9x5\frac{(3x^3+5x^2)(x^2)}{x^7-9x^5} = \frac{3x^5+5x^4}{x^7-9x^5}

(x2+x)(x2)x5βˆ’9x3=x4+x3x5βˆ’9x3\frac{(x^2+x)(x^2)}{x^5-9x^3} = \frac{x^4+x^3}{x^5-9x^3}

Combining the Numerators

Now that we have simplified the numerators, we can combine them by adding and subtracting the terms.

2x4+5x3x5βˆ’3x4βˆ’3x5+5x4x7βˆ’9x5βˆ’x4+x3x5βˆ’9x3\frac{2x^4+5x^3}{x^5-3x^4} - \frac{3x^5+5x^4}{x^7-9x^5} - \frac{x^4+x^3}{x^5-9x^3}

Finding a Common Denominator

To combine the fractions, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of x5βˆ’3x4x^5-3x^4, x7βˆ’9x5x^7-9x^5, and x5βˆ’9x3x^5-9x^3 is x7βˆ’9x5x^7-9x^5.

Rewriting the Fractions

To rewrite the fractions with the common denominator x7βˆ’9x5x^7-9x^5, we need to multiply the numerator and denominator of each fraction by the necessary factors.

2x4+5x3x5βˆ’3x4=(2x4+5x3)(x2)(x5βˆ’3x4)(x2)=(2x4+5x3)(x2)x7βˆ’3x6\frac{2x^4+5x^3}{x^5-3x^4} = \frac{(2x^4+5x^3)(x^2)}{(x^5-3x^4)(x^2)} = \frac{(2x^4+5x^3)(x^2)}{x^7-3x^6}

3x5+5x4x7βˆ’9x5=(3x5+5x4)(x2)(x7βˆ’9x5)(x2)=(3x5+5x4)(x2)x9βˆ’9x7\frac{3x^5+5x^4}{x^7-9x^5} = \frac{(3x^5+5x^4)(x^2)}{(x^7-9x^5)(x^2)} = \frac{(3x^5+5x^4)(x^2)}{x^9-9x^7}

x4+x3x5βˆ’9x3=(x4+x3)(x2)(x5βˆ’9x3)(x2)=(x4+x3)(x2)x7βˆ’9x5\frac{x^4+x^3}{x^5-9x^3} = \frac{(x^4+x^3)(x^2)}{(x^5-9x^3)(x^2)} = \frac{(x^4+x^3)(x^2)}{x^7-9x^5}

Combining the Fractions

Now that we have rewritten each fraction with the common denominator x7βˆ’9x5x^7-9x^5, we can combine them by adding and subtracting the numerators.

(2x4+5x3)(x2)x7βˆ’3x6βˆ’(3x5+5x4)(x2)x9βˆ’9x7βˆ’(x4+x3)(x2)x7βˆ’9x5\frac{(2x^4+5x^3)(x^2)}{x^7-3x^6} - \frac{(3x^5+5x^4)(x^2)}{x^9-9x^7} - \frac{(x^4+x^3)(x^2)}{x^7-9x^5}

Simplifying the Numerators

To simplify the numerators, we need to multiply the terms in each numerator.

$\frac{(2x4+5x3)(x^2)}{

Q&A: Understanding the Given Algebraic Expression

Q: What is the given algebraic expression?

A: The given algebraic expression is 2x+5x2βˆ’3xβˆ’3x+5x3βˆ’9xβˆ’x+1x2βˆ’9\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9}.

Q: What is the common denominator of the given expression?

A: The common denominator of the given expression is x3βˆ’9xx^3-9x.

Q: How do we simplify the given expression?

A: To simplify the given expression, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of x2βˆ’3xx^2-3x, x3βˆ’9xx^3-9x, and x2βˆ’9x^2-9 is x3βˆ’9xx^3-9x. We then rewrite each fraction with the common denominator and combine them by adding and subtracting the numerators.

Q: What is the simplified form of the given expression?

A: The simplified form of the given expression is (2x+5)(x)x3βˆ’3x2βˆ’(3x+5)(x2)x5βˆ’9x3βˆ’(x+1)(x)x3βˆ’9x\frac{(2x+5)(x)}{x^3-3x^2} - \frac{(3x+5)(x^2)}{x^5-9x^3} - \frac{(x+1)(x)}{x^3-9x}.

Q: How do we combine the fractions?

A: To combine the fractions, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of x3βˆ’3x2x^3-3x^2, x5βˆ’9x3x^5-9x^3, and x3βˆ’9xx^3-9x is x5βˆ’9x3x^5-9x^3. We then rewrite each fraction with the common denominator and combine them by adding and subtracting the numerators.

Q: What is the final simplified form of the given expression?

A: The final simplified form of the given expression is 2x4+5x3x7βˆ’3x6βˆ’3x5+5x4x9βˆ’9x7βˆ’x4+x3x7βˆ’9x5\frac{2x^4+5x^3}{x^7-3x^6} - \frac{3x^5+5x^4}{x^9-9x^7} - \frac{x^4+x^3}{x^7-9x^5}.

Q&A: Understanding the Options

Q: What are the options for the given expression?

A: The options for the given expression are:

A. (x+5)(x+2)x3βˆ’9x\frac{(x+5)(x+2)}{x^3-9x} B. (x+5)(x+4)x3βˆ’9x\frac{(x+5)(x+4)}{x^3-9x} C. βˆ’2x+11x3βˆ’12xβˆ’9\frac{-2x+11}{x^3-12x-9} D. 3(x+2)x2βˆ’3x\frac{3(x+2)}{x^2-3x}

Q: How do we determine the correct option?

A: To determine the correct option, we need to simplify the given expression and compare it with the options. We can do this by following the steps outlined in the previous sections.

Q: Which option is the correct answer?

A: After simplifying the given expression, we can see that the correct option is:

B. (x+5)(x+4)x3βˆ’9x\frac{(x+5)(x+4)}{x^3-9x}

Conclusion

In conclusion, the given algebraic expression 2x+5x2βˆ’3xβˆ’3x+5x3βˆ’9xβˆ’x+1x2βˆ’9\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9} can be simplified by finding a common denominator and combining the fractions. The final simplified form of the expression is 2x4+5x3x7βˆ’3x6βˆ’3x5+5x4x9βˆ’9x7βˆ’x4+x3x7βˆ’9x5\frac{2x^4+5x^3}{x^7-3x^6} - \frac{3x^5+5x^4}{x^9-9x^7} - \frac{x^4+x^3}{x^7-9x^5}. The correct option for the given expression is B. (x+5)(x+4)x3βˆ’9x\frac{(x+5)(x+4)}{x^3-9x}.