Simplify The Following Expression: (4x + 20)/(x + 4) · (3x + 12)/(6x - 12)

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Introduction

In this article, we will simplify the given algebraic expression: (4x + 20)/(x + 4) · (3x + 12)/(6x - 12). This involves applying various mathematical techniques to reduce the complexity of the expression and make it easier to understand and work with.

Understanding the Expression

The given expression is a product of two rational expressions. A rational expression is a fraction that contains variables and/or constants in the numerator and denominator. In this case, we have two rational expressions multiplied together:

(4x + 20)/(x + 4) and (3x + 12)/(6x - 12)

Step 1: Factor the Numerators and Denominators

To simplify the expression, we need to factor the numerators and denominators of both rational expressions.

Factor the Numerators

  • The numerator of the first rational expression is 4x + 20. We can factor out 4 from this expression: 4(x + 5).
  • The numerator of the second rational expression is 3x + 12. We can factor out 3 from this expression: 3(x + 4).

Factor the Denominators

  • The denominator of the first rational expression is x + 4. This is already factored.
  • The denominator of the second rational expression is 6x - 12. We can factor out 6 from this expression: 6(x - 2).

Step 2: Simplify the Expression

Now that we have factored the numerators and denominators, we can simplify the expression by canceling out any common factors between the numerators and denominators.

Cancel Out Common Factors

  • The first rational expression is (4(x + 5))/(x + 4). We can cancel out the common factor (x + 4) from the numerator and denominator.
  • The second rational expression is (3(x + 4))/(6(x - 2)). We can cancel out the common factor (x + 4) from the numerator and denominator.

Step 3: Multiply the Simplified Expressions

Now that we have simplified the individual rational expressions, we can multiply them together to get the final simplified expression.

Multiply the Simplified Expressions

  • The simplified first rational expression is 4(x + 5).
  • The simplified second rational expression is 3(x + 4)/(6(x - 2)).
  • Multiplying these two expressions together, we get: 4(x + 5) · 3(x + 4)/(6(x - 2)).

Step 4: Simplify the Final Expression

To simplify the final expression, we need to multiply the numerators and denominators together and then simplify the resulting expression.

Multiply the Numerators and Denominators

  • Multiplying the numerators together, we get: 4(x + 5) · 3(x + 4) = 12(x^2 + 9x + 20).
  • Multiplying the denominators together, we get: (x + 4) · 6(x - 2) = 6(x^2 + 2x - 8).

Step 5: Simplify Final Expression

Now that we have multiplied the numerators and denominators together, we can simplify the final expression by canceling out any common factors between the numerator and denominator.

Simplify the Final Expression

  • The final expression is 12(x^2 + 9x + 20)/(6(x^2 + 2x - 8)).
  • We can simplify this expression by canceling out the common factor 6 from the numerator and denominator.
  • The simplified final expression is 2(x^2 + 9x + 20)/(x^2 + 2x - 8).

Conclusion

In this article, we simplified the given algebraic expression: (4x + 20)/(x + 4) · (3x + 12)/(6x - 12). We applied various mathematical techniques, including factoring and canceling out common factors, to reduce the complexity of the expression and make it easier to understand and work with. The final simplified expression is 2(x^2 + 9x + 20)/(x^2 + 2x - 8).

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Introduction

In our previous article, we simplified the given algebraic expression: (4x + 20)/(x + 4) · (3x + 12)/(6x - 12). We applied various mathematical techniques, including factoring and canceling out common factors, to reduce the complexity of the expression and make it easier to understand and work with. In this article, we will answer some frequently asked questions related to the simplification of the given algebraic expression.

Q&A

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

A: The first step in simplifying the given algebraic expression is to factor the numerators and denominators of both rational expressions.

Q: How do we factor the numerators and denominators?

A: We can factor the numerators and denominators by identifying the greatest common factor (GCF) of the terms and factoring it out.

Q: What is the greatest common factor (GCF) of the terms in the numerator of the first rational expression?

A: The GCF of the terms in the numerator of the first rational expression is 4.

Q: What is the greatest common factor (GCF) of the terms in the denominator of the second rational expression?

A: The GCF of the terms in the denominator of the second rational expression is 6.

Q: How do we simplify the expression after factoring the numerators and denominators?

A: We can simplify the expression by canceling out any common factors between the numerators and denominators.

Q: What is the final simplified expression?

A: The final simplified expression is 2(x^2 + 9x + 20)/(x^2 + 2x - 8).

Q: How do we multiply the simplified expressions?

A: We can multiply the simplified expressions by multiplying the numerators and denominators together.

Q: What is the result of multiplying the simplified expressions?

A: The result of multiplying the simplified expressions is 12(x^2 + 9x + 20)/(6(x^2 + 2x - 8)).

Q: How do we simplify the final expression?

A: We can simplify the final expression by canceling out any common factors between the numerator and denominator.

Q: What is the final simplified expression?

A: The final simplified expression is 2(x^2 + 9x + 20)/(x^2 + 2x - 8).

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given algebraic expression: (4x + 20)/(x + 4) · (3x + 12)/(6x - 12). We provided step-by-step explanations and examples to help readers understand the simplification process. We hope this article has been helpful in clarifying any doubts related to the simplification of the given algebraic expression.

Additional Resources

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of various mathematical concepts, including factoring, canceling out common factors, and multiplying and simplifying expressions. In this article, we provided a step-by-step guide to simplifying the given algebraic expression: (4x + 20)/(x + 4) · (3x + 12)/(6x - 12). We hope this article has been helpful in clarifying any doubts related to the simplification of the given algebraic expression.