The Expression $\frac{6x^2 + 13x - 5}{x^2 - 16} - \frac{3x - 1}{x + 4}$ Can Be Simplified And Written In The Form $\frac{(Ax - 1)(x + B)}{(x + C)(x - D)}$, Where $A, B, C$, And $D$ Are Positive Integers.Find The

Mar 1, 2025 by ADMIN 212 views

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Problem Description

The given expression is a rational function, which is a ratio of two polynomials. The problem asks us to simplify this expression and write it in a specific form. This involves factoring the numerator and denominator, canceling out common factors, and then rewriting the expression in the desired form.

Step 1: Factor the Numerator and Denominator

To simplify the given expression, we first need to factor the numerator and denominator. The numerator is a quadratic expression, and we can factor it using the quadratic formula or by finding two numbers that multiply to give the constant term and add to give the coefficient of the linear term.

Factor the Numerator

The numerator is $6x^2 + 13x - 5$ . We can factor this expression as follows:

import sympy as sp

x = sp.symbols('x')
numerator = 6*x**2 + 13*x - 5
factored_numerator = sp.factor(numerator)
print(factored_numerator)

This will output the factored form of the numerator.

Factor the Denominator

The denominator is $x^2 - 16$ . This is a difference of squares, which can be factored as follows:

denominator = x**2 - 16
factored_denominator = sp.factor(denominator)
print(factored_denominator)

This will output the factored form of the denominator.

Step 2: Rewrite the Expression

Now that we have factored the numerator and denominator, we can rewrite the expression as follows:

\frac{(2x - 1)(3x + 5)}{(x - 4)(x + 4)} - \frac{3x - 1}{x + 4}

Step 3: Simplify the Expression

To simplify the expression, we can first cancel out the common factor of $(x + 4)$ from the first term and the second term. This gives us:

\frac{(2x - 1)(3x + 5)}{(x - 4)(x + 4)} - \frac{3x - 1}{x + 4} = \frac{(2x - 1)(3x + 5)}{(x - 4)} - 1

Step 4: Rewrite the Expression in the Desired Form

Now that we have simplified the expression, we can rewrite it in the desired form. We need to find the values of $A, B, C$ , and $D$ such that the expression can be written as:

\frac{(Ax - 1)(x + B)}{(x + C)(x - D)}

Find the Values of A, B, C, and D

To find the values of $A, B, C$ , and $D$ , we can compare the simplified expression with the desired form. We can see that:

$A = 2$
$B = 5$
$C = 4$
$D = 4$

Therefore, the expression can be written in the desired form as:

\frac{(2x - 1)(x + 5)}{(x + 4)(x - 4)}

Conclusion

In this problem, we simplified the given expression and wrote it in the desired form. We factored the numerator and denominator, canceled out common factors, and then rewrote the expression in the desired form. The final answer is $\boxed{\frac{(2x - 1)(x + 5)}{(x + 4)(x - 4)}}$ .

Final Answer

The final answer is $\boxed{\frac{(2x - 1)(x + 5)}{(x + 4)(x - 4)}}$ .

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Frequently Asked Questions

Q: What is the given expression?

A: The given expression is $\frac{6x^2 + 13x - 5}{x^2 - 16} - \frac{3x - 1}{x + 4}$ .

Q: What is the desired form of the expression?

A: The desired form of the expression is $\frac{(Ax - 1)(x + B)}{(x + C)(x - D)}$ , where $A, B, C$ , and $D$ are positive integers.

Q: How do we simplify the given expression?

A: To simplify the given expression, we need to factor the numerator and denominator, cancel out common factors, and then rewrite the expression in the desired form.

Q: What are the steps to simplify the expression?

A: The steps to simplify the expression are:

Factor the numerator and denominator.
Cancel out common factors.
Rewrite the expression in the desired form.

Q: How do we factor the numerator and denominator?

A: We can factor the numerator and denominator using the quadratic formula or by finding two numbers that multiply to give the constant term and add to give the coefficient of the linear term.

Q: What is the factored form of the numerator?

A: The factored form of the numerator is $(2x - 1)(3x + 5)$ .

Q: What is the factored form of the denominator?

A: The factored form of the denominator is $(x - 4)(x + 4)$ .

Q: How do we rewrite the expression in the desired form?

A: We can rewrite the expression in the desired form by comparing the simplified expression with the desired form and finding the values of $A, B, C$ , and $D$ .

Q: What are the values of A, B, C, and D?

A: The values of $A, B, C$ , and $D$ are:

$A = 2$
$B = 5$
$C = 4$
$D = 4$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{(2x - 1)(x + 5)}{(x + 4)(x - 4)}}$ .

Common Mistakes

Mistake 1: Not factoring the numerator and denominator

A: Failing to factor the numerator and denominator can lead to incorrect simplification of the expression.

Mistake 2: Not canceling out common factors

A: Failing to cancel out common factors can lead to incorrect simplification of the expression.

Mistake 3: Not rewriting the expression in the desired form

A: Failing to rewrite the expression in the desired form can lead to incorrect simplification of the expression.

Tips and Tricks

Tip 1: Factor the numerator and denominator carefully

A: Factoring the numerator and denominator carefully can help avoid mistakes in simplifying the expression.

Tip 2: Cancel out common factors carefully

A: Canceling out common factors carefully can help avoid mistakes in simplifying the expression.

Tip 3: Rewrite the expression in the desired form carefully

A: Rewriting the expression in the desired form carefully can help avoid mistakes in simplifying the expression.

Conclusion

In this Q&A article, we have discussed the steps to simplify the given expression and rewrite it in the desired form. We have also discussed common mistakes and tips and tricks to avoid mistakes in simplifying the expression. The final answer is $\boxed{\frac{(2x - 1)(x + 5)}{(x + 4)(x - 4)}}$ .