For What Values Of X X X Is The Rational Expression Below Undefined? Check All That Apply. X − 7 2 X 2 − 32 \frac{x-7}{2x^2-32} 2 X 2 − 32 X − 7 ​ A. -2 B. 2 C. -4 D. 4 E. 7 F. -7

Rational expressions are a fundamental concept in algebra, and understanding when they are undefined is crucial for solving equations and manipulating expressions. In this article, we will delve into the world of rational expressions and explore the values of $x$ that make the given expression undefined.

What Makes a Rational Expression Undefined?

A rational expression is undefined when the denominator is equal to zero. This is because division by zero is undefined in mathematics. Therefore, to find the values of $x$ that make the given rational expression undefined, we need to set the denominator equal to zero and solve for $x$.

The Given Rational Expression

The given rational expression is $\frac{x-7}{2x^2-32}$. To find the values of $x$ that make this expression undefined, we need to set the denominator equal to zero and solve for $x$.

Setting the Denominator Equal to Zero

The denominator of the given rational expression is $2x^2-32$. To set this expression equal to zero, we can start by factoring the quadratic expression.

import sympy as sp
x = sp.symbols('x')

denominator = 2*x**2 - 32

factored_denominator = sp.factor(denominator)
print(factored_denominator)

The factored form of the denominator is $2(x-4)(x+4)$. Now, we can set this expression equal to zero and solve for $x$.

Solving for $x$

To solve for $x$, we can set each factor equal to zero and solve for $x$.

# Set each factor equal to zero and solve for x
solutions = sp.solve(factored_denominator, x)
print(solutions)

The solutions are $x=4$ and $x=-4$. Therefore, the values of $x$ that make the given rational expression undefined are $x=4$ and $x=-4$.

Checking the Options

Now that we have found the values of $x$ that make the given rational expression undefined, we can check the options to see which ones match our solutions.

• A. -2: This value does not make the expression undefined.
• B. 2: This value does not make the expression undefined.
• C. -4: This value makes the expression undefined.
• D. 4: This value makes the expression undefined.
• E. 7: This value does not make the expression undefined.
• F. -7: This value does not make the expression undefined.

Therefore, the correct options are C and D.

Conclusion

In conclusion, the values of $x$ that make the given rational expression undefined are $x=4$ and $x=-4$. We can check the options to see which ones match our solutions and find that the correct options are C and D.

Final Answer

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In our previous article, we explored the concept of rational expressions and identified the values of $x$ that make the given expression undefined. In this article, we will continue to delve into the world of rational expressions and answer some frequently asked questions.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. It is a way of expressing a relationship between two quantities.

Q: What makes a rational expression undefined?

A: A rational expression is undefined when the denominator is equal to zero. This is because division by zero is undefined in mathematics.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression, on the other hand, is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can a rational expression have a variable in the denominator?

A: Yes, a rational expression can have a variable in the denominator. However, the variable must be raised to an even power in order for the expression to be defined.

Q: How do I add or subtract rational expressions?

A: To add or subtract rational expressions, you need to have a common denominator. You can then add or subtract the numerators while keeping the denominator the same.

Q: Can I multiply rational expressions?

A: Yes, you can multiply rational expressions. To do this, you simply multiply the numerators and denominators separately.

Q: What is the rule for multiplying rational expressions?

A: The rule for multiplying rational expressions is that you multiply the numerators and denominators separately, and then simplify the resulting expression.

Q: Can I divide rational expressions?

A: Yes, you can divide rational expressions. To do this, you can multiply the numerator and denominator by the reciprocal of the divisor.

Q: What is the rule for dividing rational expressions?

A: The rule for dividing rational expressions is that you multiply the numerator and denominator by the reciprocal of the divisor, and then simplify the resulting expression.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, the variable must be raised to an even power in order for the expression to be defined.

Q: How do I determine if a rational expression is undefined?

A: To determine if a rational expression is undefined, you need to check if the denominator is equal to zero. If the denominator is equal to zero, then the expression is undefined.

Q: Can a rational expression have a negative exponent?

A: Yes, a rational expression can have a negative exponent. However, the negative exponent must be raised to an even power in order for the expression to be defined.

Q: How do I simplify a rational expression with a negative exponent?

A: To simplify a rational expression with a negative exponent, you need to raise the base to the power of the absolute value of the exponent, and then change the sign of the exponent.

Conclusion

In conclusion, rational expressions are a fundamental concept in algebra, and understanding how to simplify and manipulate them is crucial for solving equations and manipulating expressions. We hope that this Q&A guide has helped to clarify any questions you may have had about rational expressions.

Final Answer

The final answer is that rational expressions are a powerful tool for solving equations and manipulating expressions, and understanding how to simplify and manipulate them is crucial for success in algebra.