Simplify The Expression: 2 X + 8 X 2 + 4 X + 3 + 1 X + 3 \frac{2x+8}{x^2+4x+3}+\frac{1}{x+3} X 2 + 4 X + 3 2 X + 8 ​ + X + 3 1 ​

by ADMIN 129 views

Introduction

In this article, we will simplify the given expression 2x+8x2+4x+3+1x+3\frac{2x+8}{x^2+4x+3}+\frac{1}{x+3}. This involves combining the two fractions and simplifying the resulting expression. We will use various algebraic techniques, including factoring and canceling common factors, to simplify the expression.

Step 1: Factor the Denominator of the First Fraction

The first step in simplifying the expression is to factor the denominator of the first fraction, which is x2+4x+3x^2+4x+3. We can factor this quadratic expression as (x+1)(x+3)(x+1)(x+3).

import sympy as sp

x = sp.symbols('x')



denominator = x**2 + 4*x + 3
factored_denominator = sp.factor(denominator)


print(factored_denominator)

Step 2: Rewrite the Expression with the Factored Denominator

Now that we have factored the denominator of the first fraction, we can rewrite the expression with the factored denominator.

2x+8(x+1)(x+3)+1x+3\frac{2x+8}{(x+1)(x+3)}+\frac{1}{x+3}

Step 3: Factor the Numerator of the First Fraction

The next step is to factor the numerator of the first fraction, which is 2x+82x+8. We can factor this linear expression as 2(x+4)2(x+4).

# Factor the numerator of the first fraction
numerator = 2*x + 8
factored_numerator = sp.factor(numerator)

print(factored_numerator)

Step 4: Rewrite the Expression with the Factored Numerator

Now that we have factored the numerator of the first fraction, we can rewrite the expression with the factored numerator.

2(x+4)(x+1)(x+3)+1x+3\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{x+3}

Step 5: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1x+3=2(x+4)(x+1)(x+3)+(x+3)(x+3)(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{x+3} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{(x+3)}{(x+3)(x+3)}

Step 6: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+(x+3)(x+3)(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{(x+3)}{(x+3)(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 7: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 8: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 9: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 10: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 11: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 12: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 13: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 14: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 15: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 16: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 17: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 18: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the numerator and denominator of the second fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

2(x+4)(x+1)(x+3)+1(x+3)=2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)} = \frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}

Step 19: Simplify the Expression by Canceling Common Factors

We can simplify the expression by canceling common factors between the

Introduction

In our previous article, we simplified the expression 2x+8x2+4x+3+1x+3\frac{2x+8}{x^2+4x+3}+\frac{1}{x+3} by factoring the denominator and canceling common factors. In this article, we will answer some frequently asked questions about the simplification process.

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

A: The first step in simplifying the expression is to factor the denominator of the first fraction, which is x2+4x+3x^2+4x+3. We can factor this quadratic expression as (x+1)(x+3)(x+1)(x+3).

Q: Why do we need to factor the denominator?

A: We need to factor the denominator to simplify the expression and make it easier to work with. Factoring the denominator allows us to cancel common factors between the numerator and denominator.

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to factor the numerator of the first fraction, which is 2x+82x+8. We can factor this linear expression as 2(x+4)2(x+4).

Q: Why do we need to factor the numerator?

A: We need to factor the numerator to simplify the expression and make it easier to work with. Factoring the numerator allows us to cancel common factors between the numerator and denominator.

Q: What is the final simplified expression?

A: The final simplified expression is 2(x+4)(x+1)(x+3)+1(x+3)\frac{2(x+4)}{(x+1)(x+3)}+\frac{1}{(x+3)}.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by canceling common factors between the numerator and denominator of the first fraction. In this case, we can cancel the factor (x+3)(x+3) from the numerator and denominator.

Q: What is the final simplified expression after canceling common factors?

A: The final simplified expression after canceling common factors is 2(x+4)(x+1)\frac{2(x+4)}{(x+1)}.

Q: Is the expression simplified?

A: Yes, the expression is simplified.

Q: Can we simplify the expression further?

A: No, the expression cannot be simplified further.

Q: What is the final answer?

A: The final answer is 2(x+4)(x+1)\frac{2(x+4)}{(x+1)}.

Conclusion

In this article, we answered some frequently asked questions about the simplification process of the expression 2x+8x2+4x+3+1x+3\frac{2x+8}{x^2+4x+3}+\frac{1}{x+3}. We covered the first step in simplifying the expression, factoring the denominator, factoring the numerator, canceling common factors, and the final simplified expression.

Additional Resources

Related Articles