Simplify. Express Your Answer As A Single Fraction In Simplest Form. 4 D + 4 D 2 − 4 D − 32 + D D + 4 \frac{4d + 4}{d^2 - 4d - 32} + \frac{d}{d + 4} D 2 − 4 D − 32 4 D + 4 ​ + D + 4 D ​

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Introduction

In this article, we will simplify the given algebraic expression, which involves adding two fractions with different denominators. The expression is 4d+4d2−4d−32+dd+4\frac{4d + 4}{d^2 - 4d - 32} + \frac{d}{d + 4}. We will use various algebraic techniques to simplify the expression and express it as a single fraction in its simplest form.

Step 1: Factor the Denominators

The first step in simplifying the given expression is to factor the denominators of both fractions. The denominator of the first fraction is d2−4d−32d^2 - 4d - 32, which can be factored as (d−8)(d+4)(d - 8)(d + 4). The denominator of the second fraction is d+4d + 4, which is already factored.

\frac{4d + 4}{(d - 8)(d + 4)} + \frac{d}{d + 4}

Step 2: Simplify the First Fraction

To simplify the first fraction, we can factor the numerator as 4(d+1)4(d + 1). This allows us to rewrite the fraction as 4(d+1)(d−8)(d+4)\frac{4(d + 1)}{(d - 8)(d + 4)}.

\frac{4(d + 1)}{(d - 8)(d + 4)} + \frac{d}{d + 4}

Step 3: Simplify the Second Fraction

The second fraction can be simplified by canceling out the common factor of d+4d + 4 in the numerator and denominator. This leaves us with d1\frac{d}{1}, which simplifies to dd.

\frac{4(d + 1)}{(d - 8)(d + 4)} + d

Step 4: Find a Common Denominator

To add the two fractions, we need to find a common denominator. The common denominator is (d−8)(d+4)(d - 8)(d + 4).

\frac{4(d + 1)}{(d - 8)(d + 4)} + \frac{d(d - 8)(d + 4)}{(d - 8)(d + 4)}

Step 5: Add the Fractions

Now that we have a common denominator, we can add the two fractions. This involves adding the numerators and keeping the common denominator.

\frac{4(d + 1) + d(d - 8)(d + 4)}{(d - 8)(d + 4)}

Step 6: Simplify the Numerator

To simplify the numerator, we can expand the expression d(d−8)(d+4)d(d - 8)(d + 4) and combine like terms.

\frac{4(d + 1) + d(d^2 - 4d - 32)}{(d - 8)(d + 4)}

Step 7: Expand and Combine Like Terms

Expanding the expression d(d2−4d−32)d(d^2 - 4d - 32) gives us d3−4d2−32dd^3 - 4d^2 - 32d. We can now combine like terms in the numerator.

\frac{4d + 4 + d^3 - 4d^2 - 32d}{(d - 8)(d + 4)}

Step 8: Simplify the Numerator

Combining like terms in the numerator gives us d3−4d2−28d+4d^3 - 4d^2 - 28d + 4.

\frac{d^3 - 4d^2 - 28d + 4}{(d - 8)(d + 4)}

Step 9: Factor the Numerator

The numerator can be factored as (d−8)(d2+4d+1)+4(d - 8)(d^2 + 4d + 1) + 4.

\frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)(d + 4)}

Step 10: Simplify the Expression

We can now simplify the expression by canceling out the common factor of (d−8)(d - 8) in the numerator and denominator.

\frac{d^2 + 4d + 1 + \frac{4}{d - 8}}{d + 4}

Step 11: Simplify the Expression Further

The expression can be simplified further by combining the terms in the numerator.

\frac{d^2 + 4d + 1 + \frac{4}{d - 8}}{d + 4} = \frac{(d + 4)(d^2 + 4d + 1) + 4}{(d + 4)(d - 8)}

Step 12: Simplify the Expression Even Further

We can now simplify the expression even further by canceling out the common factor of (d+4)(d + 4) in the numerator and denominator.

\frac{(d + 4)(d^2 + 4d + 1) + 4}{(d + 4)(d - 8)} = \frac{d^2 + 4d + 1 + \frac{4}{d - 8}}{d - 8}

Step 13: Simplify the Expression to Its Final Form

The expression can be simplified to its final form by combining the terms in the numerator.

\frac{d^2 + 4d + 1 + \frac{4}{d - 8}}{d - 8} = \frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)^2}

Conclusion

In this article, we simplified the given algebraic expression, which involved adding two fractions with different denominators. We used various algebraic techniques to simplify the expression and express it as a single fraction in its simplest form. The final simplified expression is (d−8)(d2+4d+1)+4(d−8)2\frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)^2}.

Final Answer

The final answer is (d−8)(d2+4d+1)+4(d−8)2\boxed{\frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)^2}}.

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Introduction

In our previous article, we simplified the given algebraic expression, which involved adding two fractions with different denominators. We used various algebraic techniques to simplify the expression and express it as a single fraction in its simplest form. In this article, we will answer some frequently asked questions related to the simplification of the given expression.

Q&A

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

A: The final simplified expression is (d−8)(d2+4d+1)+4(d−8)2\frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)^2}.

Q: How did you simplify the expression?

A: We used various algebraic techniques, including factoring, expanding, and combining like terms, to simplify the expression.

Q: What is the common denominator of the two fractions?

A: The common denominator is (d−8)(d+4)(d - 8)(d + 4).

Q: How did you find the common denominator?

A: We factored the denominators of both fractions and found the least common multiple (LCM) of the two factors.

Q: What is the numerator of the simplified expression?

A: The numerator is (d−8)(d2+4d+1)+4(d - 8)(d^2 + 4d + 1) + 4.

Q: How did you simplify the numerator?

A: We expanded the expression d(d2−4d−32)d(d^2 - 4d - 32) and combined like terms.

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

A: The final form of the simplified expression is (d−8)(d2+4d+1)+4(d−8)2\frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)^2}.

Common Mistakes

Mistake 1: Not factoring the denominators

  • Not factoring the denominators can make it difficult to find the common denominator and simplify the expression.
  • To avoid this mistake, make sure to factor the denominators of both fractions.

Mistake 2: Not expanding and combining like terms

  • Not expanding and combining like terms can make it difficult to simplify the numerator and find the final form of the expression.
  • To avoid this mistake, make sure to expand and combine like terms in the numerator.

Mistake 3: Not canceling out common factors

  • Not canceling out common factors can make the expression more complicated than it needs to be.
  • To avoid this mistake, make sure to cancel out common factors in the numerator and denominator.

Tips and Tricks

Tip 1: Factor the denominators first

  • Factoring the denominators first can make it easier to find the common denominator and simplify the expression.
  • To do this, factor the denominators of both fractions and find the least common multiple (LCM) of the two factors.

Tip 2: Expand and combine like terms

  • Expanding and combining like terms can make it easier to simplify the numerator and find the final form of the expression.
  • To do this, expand the expression and combine like terms in the numerator.

Tip 3: Cancel out common factors

  • Canceling out common factors can make the expression simpler and easier to understand.
  • To do this, cancel out common factors in the numerator and denominator.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given algebraic expression. We also discussed some common mistakes to avoid and provided some tips and tricks to make the simplification process easier. By following these tips and tricks, you can simplify the given expression and express it as a single fraction in its simplest form.

Final Answer

The final answer is (d−8)(d2+4d+1)+4(d−8)2\boxed{\frac{(d - 8)(d^2 + 4d + 1) + 4}{(d - 8)^2}}.