What Is The Sum?$\[ \frac{3}{x^2-9}+\frac{5}{x+3} \\]$\[ \frac{8}{x^2+x-6} \\]$\[ \frac{5x-12}{x-3} \\]$\[ \frac{-5x}{(x+3)(x-3)} \\]$\[ \frac{5x-12}{(x+3)(x-3)} \\]

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Introduction

When dealing with fractions, especially those with different denominators, it can be challenging to find their sum. In this article, we will explore the sum of the given fractions, which involves combining multiple fractions with different denominators. We will use algebraic techniques to simplify the expression and find the final result.

The Given Fractions

The given fractions are:

3x2−9+5x+3\frac{3}{x^2-9}+\frac{5}{x+3}

8x2+x−6\frac{8}{x^2+x-6}

5x−12x−3\frac{5x-12}{x-3}

−5x(x+3)(x−3)\frac{-5x}{(x+3)(x-3)}

5x−12(x+3)(x−3)\frac{5x-12}{(x+3)(x-3)}

Step 1: Factor the Denominators

To find the sum of the fractions, we need to factor the denominators of each fraction. This will help us identify any common factors that can be canceled out.

The first fraction has a denominator of x2−9x^2-9, which can be factored as (x−3)(x+3)(x-3)(x+3).

The second fraction has a denominator of x2+x−6x^2+x-6, which can be factored as (x−2)(x+3)(x-2)(x+3).

The third fraction has a denominator of x−3x-3, which is already factored.

The fourth fraction has a denominator of (x+3)(x−3)(x+3)(x-3), which is already factored.

The fifth fraction also has a denominator of (x+3)(x−3)(x+3)(x-3), which is already factored.

Step 2: Rewrite the Fractions with Common Denominators

Now that we have factored the denominators, we can rewrite each fraction with a common denominator of (x+3)(x−3)(x+3)(x-3).

3(x−3)(x+3)+5(x+3)\frac{3}{(x-3)(x+3)}+\frac{5}{(x+3)}

8(x−2)(x+3)\frac{8}{(x-2)(x+3)}

5x−12x−3\frac{5x-12}{x-3}

−5x(x+3)(x−3)\frac{-5x}{(x+3)(x-3)}

5x−12(x+3)(x−3)\frac{5x-12}{(x+3)(x-3)}

Step 3: Find the Least Common Multiple (LCM)

To add the fractions, we need to find the least common multiple (LCM) of the denominators. In this case, the LCM is (x+3)(x−3)(x+3)(x-3).

Step 4: Rewrite the Fractions with the LCM

Now that we have found the LCM, we can rewrite each fraction with the LCM as the denominator.

3(x+3)(x+3)(x+3)+5(x−3)(x+3)(x−3)\frac{3(x+3)}{(x+3)(x+3)}+\frac{5(x-3)}{(x+3)(x-3)}

8(x−2)(x−2)(x+3)\frac{8(x-2)}{(x-2)(x+3)}

(5x−12)(x+3)(x−3)(x+3)\frac{(5x-12)(x+3)}{(x-3)(x+3)}

−5x(x−3)(x+3)(x−3)\frac{-5x(x-3)}{(x+3)(x-3)}

(5x−12)(x+3)(x+3)(x−3)\frac{(5x-12)(x+3)}{(x+3)(x-3)}

Step 5: Add the Fractions

Now that we have rewritten each fraction with the LCM as the denominator, we can add the fractions.

3(x+3)(x+3)(x+3)+5(x−3)(x+3)(x−3)+8(x−2)(x−2)(x+3)+(5x−12)(x+3)(x−3)(x+3)+−5x(x−3)(x+3)(x−3)+(5x−12)(x+3)(x+3)(x−3)\frac{3(x+3)}{(x+3)(x+3)}+\frac{5(x-3)}{(x+3)(x-3)}+\frac{8(x-2)}{(x-2)(x+3)}+\frac{(5x-12)(x+3)}{(x-3)(x+3)}+\frac{-5x(x-3)}{(x+3)(x-3)}+\frac{(5x-12)(x+3)}{(x+3)(x-3)}

Step 6: Simplify the Expression

To simplify the expression, we can cancel out any common factors in the numerator and denominator.

3(x+3)(x+3)(x+3)+5(x−3)(x+3)(x−3)+8(x−2)(x−2)(x+3)+(5x−12)(x+3)(x−3)(x+3)+−5x(x−3)(x+3)(x−3)+(5x−12)(x+3)(x+3)(x−3)\frac{3(x+3)}{(x+3)(x+3)}+\frac{5(x-3)}{(x+3)(x-3)}+\frac{8(x-2)}{(x-2)(x+3)}+\frac{(5x-12)(x+3)}{(x-3)(x+3)}+\frac{-5x(x-3)}{(x+3)(x-3)}+\frac{(5x-12)(x+3)}{(x+3)(x-3)}

=3(x+3)(x+3)(x+3)+5(x−3)(x+3)(x−3)+8(x−2)(x−2)(x+3)+(5x−12)(x+3)(x−3)(x+3)+−5x(x−3)(x+3)(x−3)+(5x−12)(x+3)(x+3)(x−3)=\frac{3(x+3)}{(x+3)(x+3)}+\frac{5(x-3)}{(x+3)(x-3)}+\frac{8(x-2)}{(x-2)(x+3)}+\frac{(5x-12)(x+3)}{(x-3)(x+3)}+\frac{-5x(x-3)}{(x+3)(x-3)}+\frac{(5x-12)(x+3)}{(x+3)(x-3)}

=3x+3+5x−3+8x−2+5x−12x−3+−5xx−3+5x−12x−3=\frac{3}{x+3}+\frac{5}{x-3}+\frac{8}{x-2}+\frac{5x-12}{x-3}+\frac{-5x}{x-3}+\frac{5x-12}{x-3}

Step 7: Combine Like Terms

Now that we have simplified the expression, we can combine like terms.

=3x+3+5x−3+8x−2+5x−12x−3+−5xx−3+5x−12x−3=\frac{3}{x+3}+\frac{5}{x-3}+\frac{8}{x-2}+\frac{5x-12}{x-3}+\frac{-5x}{x-3}+\frac{5x-12}{x-3}

=3x+3+5x−3+8x−2+5x−12−5xx−3+5x−12x−3=\frac{3}{x+3}+\frac{5}{x-3}+\frac{8}{x-2}+\frac{5x-12-5x}{x-3}+\frac{5x-12}{x-3}

=3x+3+5x−3+8x−2+−12x−3+5x−12x−3=\frac{3}{x+3}+\frac{5}{x-3}+\frac{8}{x-2}+\frac{-12}{x-3}+\frac{5x-12}{x-3}

Step 8: Simplify Further

Now that we have combined like terms, we can simplify further.

=3x+3+5x−3+8x−2+−12x−3+5x−12x−3=\frac{3}{x+3}+\frac{5}{x-3}+\frac{8}{x-2}+\frac{-12}{x-3}+\frac{5x-12}{x-3}

=3x+3+5−12+5x−12x−3+8x−2=\frac{3}{x+3}+\frac{5-12+5x-12}{x-3}+\frac{8}{x-2}

=3x+3+5x−35x−3+8x−2=\frac{3}{x+3}+\frac{5x-35}{x-3}+\frac{8}{x-2}

Step 9: Find the Final Result

Now that we have simplified the expression, we can find the final result.

=3x+3+5x−35x−3+8x−2=\frac{3}{x+3}+\frac{5x-35}{x-3}+\frac{8}{x-2}

=3(x−3)(x+3)(x−3)+(5x−35)(x+3)(x−3)(x+3)+8(x+3)(x−2)(x+3)=\frac{3(x-3)}{(x+3)(x-3)}+\frac{(5x-35)(x+3)}{(x-3)(x+3)}+\frac{8(x+3)}{(x-2)(x+3)}

=3x−9(x+3)(x−3)+5x2−35x−105(x−3)(x+3)+8x+24(x−2)(x+3)=\frac{3x-9}{(x+3)(x-3)}+\frac{5x^2-35x-105}{(x-3)(x+3)}+\frac{8x+24}{(x-2)(x+3)}

=(3x−9)(x+3)+(5x2−35x−105)(x+3)(x−3)+8x+24(x−2)(x+3)=\frac{(3x-9)(x+3)+(5x^2-35x-105)}{(x+3)(x-3)}+\frac{8x+24}{(x-2)(x+3)}

=3x2−27x−27+5x2−35x−105(x+3)(x−3)+8x+24(x−2)(x+3)=\frac{3x^2-27x-27+5x^2-35x-105}{(x+3)(x-3)}+\frac{8x+24}{(x-2)(x+3)}

=8x2−62x−132(x+3)(x−3)+8x+24(x−2)(x+3)=\frac{8x^2-62x-132}{(x+3)(x-3)}+\frac{8x+24}{(x-2)(x+3)}

=(8x2−62x−132)(x−2)+(8x+24)(x+3)(x+3)(x−3)(x−2)=\frac{(8x^2-62x-132)(x-2)+(8x+24)(x+3)}{(x+3)(x-3)(x-2)}

=8x3−64x2−132x−2x2+124x+264(x+3)(x−3)(x−2)=\frac{8x^3-64x^2-132x-2x^2+124x+264}{(x+3)(x-3)(x-2)}

=8x3−66x2−8x+264(x+3)(x−3)(x−2)=\frac{8x^3-66x^2-8x+264}{(x+3)(x-3)(x-2)}

Conclusion

In this article, we have explored the sum of the given fractions, which involves combining multiple fractions with different denominators. We have used algebraic techniques to simplify the expression and find the final result. The final result is 8x3−66x2−8x+264(x+3)(x−3)(x−2)\frac{8x^3-66x^2-8x+264}{(x+3)(x-3)(x-2)}.

Q: What is the sum of the given fractions?

A: The sum of the given fractions is 8x3−66x2−8x+264(x+3)(x−3)(x−2)\frac{8x^3-66x^2-8x+264}{(x+3)(x-3)(x-2)}.

Q: How do I simplify the expression?

A: To simplify the expression, you can use algebraic techniques such as factoring, canceling out common factors, and combining like terms.

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is (x+3)(x−3)(x+3)(x-3).

Q: How do I find the LCM?

A: To find the LCM, you can list the factors of each denominator and find the product of the highest powers of each factor.

Q: What is the final result?

A: The final result is 8x3−66x2−8x+264(x+3)(x−3)(x−2)\frac{8x^3-66x^2-8x+264}{(x+3)(x-3)(x-2)}.

Q: How do I simplify the expression further?

A: To simplify the expression further, you can cancel out any common factors in the numerator and denominator.

Q: What is the difference between the numerator and denominator?

A: The difference between the numerator and denominator is 8x3−66x2−8x+2648x^3-66x^2-8x+264 and (x+3)(x−3)(x−2)(x+3)(x-3)(x-2).

Q: How do I factor the numerator?

A: To factor the numerator, you can look for common factors and use algebraic techniques such as factoring by grouping.

Q: What is the final simplified expression?

A: The final simplified expression is 8x3−66x2−8x+264(x+3)(x−3)(x−2)\frac{8x^3-66x^2-8x+264}{(x+3)(x-3)(x-2)}.

Q: How do I check my work?

A: To check your work, you can plug in values for x and simplify the expression to see if it matches the final result.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Not canceling out common factors
  • Not combining like terms
  • Not factoring the numerator and denominator correctly
  • Not checking your work

Q: How do I use algebraic techniques to simplify expressions?

A: To use algebraic techniques to simplify expressions, you can:

  • Factor the numerator and denominator
  • Cancel out common factors
  • Combine like terms
  • Use algebraic identities to simplify the expression

Q: What are some real-world applications of simplifying expressions?

A: Some real-world applications of simplifying expressions include:

  • Physics: Simplifying expressions is used to solve problems involving motion, energy, and forces.
  • Engineering: Simplifying expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Simplifying expressions is used to model and analyze economic systems, such as supply and demand curves.

Q: How do I apply algebraic techniques to real-world problems?

A: To apply algebraic techniques to real-world problems, you can:

  • Identify the variables and constants in the problem
  • Use algebraic techniques to simplify the expression
  • Use the simplified expression to solve the problem

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

  • Start by simplifying the numerator and denominator separately
  • Use algebraic techniques such as factoring and canceling out common factors
  • Check your work by plugging in values for x and simplifying the expression
  • Use algebraic identities to simplify the expression

Q: How do I know if I have simplified the expression correctly?

A: To know if you have simplified the expression correctly, you can:

  • Check your work by plugging in values for x and simplifying the expression
  • Use algebraic techniques to simplify the expression
  • Compare your simplified expression to the original expression to see if it matches.

Q: What are some common algebraic identities that can be used to simplify expressions?

A: Some common algebraic identities that can be used to simplify expressions include:

  • (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2
  • (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2
  • a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
  • a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Q: How do I use algebraic identities to simplify expressions?

A: To use algebraic identities to simplify expressions, you can:

  • Identify the algebraic identity that matches the expression
  • Use the algebraic identity to simplify the expression
  • Check your work by plugging in values for x and simplifying the expression.