Express in simplest form: 3 x+184 x+8÷x2−36x+2.(1 point) Responses 14(x−2) Start Fraction 1 over 4 left parenthesis x minus 2 right parenthesis End Fraction 4(x−6)3 Start Fraction 4 left parenthesis x minus 6 right parenthesis over 3 Endfraction 34x−24 Start Fraction 2 over 4 x minus 24 End Fraction 34x−8

Answer 1

To simplify the expression \( 3x + \frac{184x + 8}{x^2 - 36x + 2} \), we start with the denominator and factor it.

1. Factor the quadratic: The expression \( x^2 - 36x + 2 \) does not factor easily (as it has complex roots). So we will leave it as is.

2. Substitute and simplify: We can rewrite the expression as follows: \[ 3x + \frac{184x + 8}{x^2 - 36x + 2} \]

3. Combine fractions: To combine the two terms, we express \( 3x \) as a fraction with the same denominator: \[ 3x = \frac{3x(x^2 - 36x + 2)}{x^2 - 36x + 2} \] So the expression becomes: \[ \frac{3x(x^2 - 36x + 2) + 184x + 8}{x^2 - 36x + 2} \]

4. Multiply out the numerator: Now we expand the numerator: \[ 3x(x^2 - 36x + 2) = 3x^3 - 108x^2 + 6x \] Then we combine like terms: \[ 3x^3 - 108x^2 + 6x + 184x + 8 = 3x^3 - 108x^2 + 190x + 8 \] So now, our expression is: \[ \frac{3x^3 - 108x^2 + 190x + 8}{x^2 - 36x + 2} \]

5. Look for further simplification: Since the numerator does not seem to have common factors with the denominator, we have: \[ = \frac{3x^3 - 108x^2 + 190x + 8}{x^2 - 36x + 2} \] This cannot be simplified further without additional context or specific values.

From the original responses, we would check whether \( 3x + \frac{184x + 8}{x^2 - 36x + 2} \) somehow simplifies to any of those forms given in the answers. It doesn't clearly match any directly.

If you have any other conditions or specific substitutions, please share, and I can assist you further!

Answer 2

So what is the answer

Answer 3

Let's go through the original expression step by step to find the answer.

The original expression is:

\[ 3x + \frac{184x + 8}{x^2 - 36x + 2} \]

1. Combine the terms: We want to combine \( 3x \) and the fraction. First, rewrite \( 3x \) with the same denominator:

   \[ 3x = \frac{3x(x^2 - 36x + 2)}{x^2 - 36x + 2} = \frac{3x^3 - 108x^2 + 6x}{x^2 - 36x + 2} \]

   Now we can combine the two parts:

   \[ \frac{3x^3 - 108x^2 + 6x + 184x + 8}{x^2 - 36x + 2} \]

   This simplifies to:

   \[ \frac{3x^3 - 108x^2 + 190x + 8}{x^2 - 36x + 2} \]

2. Check if we can factor the numerator: Unfortunately, factoring \( 3x^3 - 108x^2 + 190x + 8 \) may not yield nice factors, and it doesn't look like we can simplify it any further without specific values of \( x \).

Now, let's simplify the given options based on \( x^2 - 36x + 2 \):

1. The denominator \( x^2 - 36x + 2 \) does not factor simply, so we cannot use the responses directly.

Given response choices:

• \( 14(x - 2) \)
• \( \frac{1}{4}(x - 2) \)
• \( 4(x - 6) \)
• \( \frac{4(x - 6)}{3} \)
• \( \frac{2}{4x - 24} \)
• \( 34x - 8 \)

None of these directly appears as a simplification of the expression we derived.

Conclusion: Since the expression simplifies to a rational function and none of the choices provided directly match, it appears none of the options given present a valid simplification of the original expression. However, the expanded expression is the most compact version as per standard mathematical notation.

If you're looking for a numerical value or a particular condition of \( x \) that satisfies any options, please clarify that! Otherwise, please cross-check if any specific conditions or restrictions apply to reduce the original expression differently.

Answer 4

Express in simplest form: 3x+18 over 4x+8 divided by x^3-36 over x+2

Answer 5

To simplify the given expression

\[ \frac{3x + 18}{4x + 8} \div \frac{x^3 - 36}{x + 2}, \]

we'll follow these steps:

Step 1: Rewrite the Division as Multiplication

We can rewrite the division as multiplication by inverting the second fraction:

\[ \frac{3x + 18}{4x + 8} \times \frac{x + 2}{x^3 - 36}. \]

Step 2: Factor Each Component

Now we will factor where possible.

For the first fraction:

• \(3x + 18 = 3(x + 6)\),
• \(4x + 8 = 4(x + 2)\).

So we can rewrite the first fraction:

\[ \frac{3(x + 6)}{4(x + 2)}. \]

For the second fraction:

• The numerator \( x^3 - 36 \) can be factored as a difference of cubes:
  • \( x^3 - 36 = (x - 6)(x^2 + 6x + 36) \).
• The denominator is simply \( x + 2 \).

So we can rewrite the second fraction as:

\[ \frac{x + 2}{(x - 6)(x^2 + 6x + 36)}. \]

Step 3: Combine the Fractions

Now we can combine the two fractions:

\[ \frac{3(x + 6)}{4(x + 2)} \times \frac{x + 2}{(x - 6)(x^2 + 6x + 36)}. \]

Next, the \( (x + 2) \) in the numerator of the second fraction and the denominator of the first fraction cancels out:

\[ = \frac{3(x + 6)}{4} \times \frac{1}{(x - 6)(x^2 + 6x + 36)}. \]

Step 4: Simplify the Expression

So we are left with:

\[ \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)}. \]

Final Answer

Thus, the expression in its simplest form is:

\[ \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)}. \]

Answer 6

Not an option

Answer 7

I apologize for any confusion. Let's approach the expression step by step again to ensure we simplify correctly and check against typical options or simplifications.

Given expression:

\[ \frac{3x + 18}{4x + 8} \div \frac{x^3 - 36}{x + 2} \]

Step 1: Rewrite the Division as Multiplication

We rewrite the division as multiplication by the reciprocal:

\[ \frac{3x + 18}{4x + 8} \times \frac{x + 2}{x^3 - 36} \]

Step 2: Factor Each Term

1. Factor \(3x + 18\):

   \[ 3x + 18 = 3(x + 6) \]

2. Factor \(4x + 8\):

   \[ 4x + 8 = 4(x + 2) \]

3. Factor \(x^3 - 36\):

   \(x^3 - 36\) can be factored as \(x^3 - 6^2\), which is a difference of squares, further factoring gives:

   \[ x^3 - 36 = (x - 6)(x^2 + 6x + 36) \]

Step 3: Substitute the Factors into the Expression

Now we substitute the factors back:

\[ \frac{3(x + 6)}{4(x + 2)} \times \frac{x + 2}{(x - 6)(x^2 + 6x + 36)} \]

Step 4: Cancel Common Terms

Notice that \(x + 2\) in the numerator and denominator will cancel:

\[ = \frac{3(x + 6)}{4} \times \frac{1}{(x - 6)(x^2 + 6x + 36)} \]

This simplifies to:

\[ = \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)} \]

Final Simplification

At this stage, we confirm if this is already in the simplest form.

1. There are no further common factors between the numerator and denominator at this point.
2. We cannot simplify \(x^2 + 6x + 36\) easily, and it does not factor into real numbers.

Alternative Simplification Check

Let’s check if there's any numerical evaluation based on possible values or checking any factors if we made a simplification error at the start.

Review of common question options:

If the expected answer was indeed among options provided, can you provide those options again?

Otherwise, if we eliminate or further check corners for \(x \neq -2\) and \(x \neq 6\), the simplest form remains:

\[ \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)} \]

Please let me know if there are specific options you see or further details to adjust, and I will assist!

Answer 8

Can you format the fraction like I did I dont understand

Answer 9

Absolutely! Let's format it step by step in the same style as your original question.

We want to simplify the following expression:

\[ \frac{3x + 18}{4x + 8} \div \frac{x^3 - 36}{x + 2} \]

Step 1: Rewrite the Division as Multiplication

We can rewrite the division as multiplication by flipping the second fraction:

\[ \frac{3x + 18}{4x + 8} \times \frac{x + 2}{x^3 - 36} \]

Step 2: Factor Each Part

For the first fraction:

1. \(3x + 18\) can be factored as: \[ 3(x + 6) \]

2. \(4x + 8\) can be factored as: \[ 4(x + 2) \]

For the second fraction:

1. \(x^3 - 36\) can be factored as a difference of cubes: \[ x^3 - 36 = (x - 6)(x^2 + 6x + 36) \] Note: \(36\) is treated as \(6^2\) here in the context.

Step 3: Substitute the Factored Forms

Now we substitute back these factorizations into our expression:

\[ \frac{3(x + 6)}{4(x + 2)} \times \frac{x + 2}{(x - 6)(x^2 + 6x + 36)} \]

Step 4: Cancel Common Factors

In this case, we can see that \( (x + 2) \) appears in both the numerator and the denominator, allowing us to cancel it:

\[ = \frac{3(x + 6)}{4} \times \frac{1}{(x - 6)(x^2 + 6x + 36)} \]

Final Form

After cancelling, we arrive at the final simplified expression:

\[ = \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)} \]

So, in this format:

\[ \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)} \]

This is the simplest form of the original expression. If you need any further assistance or explanations, feel free to ask!

Answer 10

Not an option, I think you need to do the problem again it seems you are gueesing

Answer 11

I apologize for any confusion earlier. Let’s go through the problem step by step clearly and ensure we arrive at a simplified final answer that matches common formats.

We start with:

\[ \frac{3x + 18}{4x + 8} \div \frac{x^3 - 36}{x + 2} \]

Step 1: Rewrite the Division as Multiplication

We change the division into multiplication by flipping the second fraction:

\[ \frac{3x + 18}{4x + 8} \times \frac{x + 2}{x^3 - 36} \]

Step 2: Factor Each Part

1. Factor \(3x + 18\): \[ 3x + 18 = 3(x + 6) \]

2. Factor \(4x + 8\): \[ 4x + 8 = 4(x + 2) \]

3. Factor \(x^3 - 36\): The expression \(x^3 - 36\) can be written as \(x^3 - (6^2)\). This is a difference of squares: \[ x^3 - 36 = (x - 6)(x^2 + 6x + 36) \] (Note: This was misrepresented before, as it doesn't factor further simply; we'll leave it like this.)

Step 3: Substitute the Factors

Now substituting the factors back, we have:

\[ \frac{3(x + 6)}{4(x + 2)} \times \frac{x + 2}{(x - 6)(x^2 + 6x + 36)} \]

Step 4: Cancel Common Factors

Now we can cancel \( (x + 2) \) from the numerator and denominator:

\[ = \frac{3(x + 6)}{4} \times \frac{1}{(x - 6)(x^2 + 6x + 36)} \]

This becomes:

\[ = \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)} \]

Final Simplification

This simplification is already in a manageable form.

However, if we look for a numerical evaluation or something more basic:

Let’s check values of certain limits within the context of \(x\) for simpler values.

But if you want to check if this could yield various simpler more common responses such as canceling down:

Since we already canceled down the expression effectively, let’s check if \(x\) values like specific numbers yield common factorization.

Ultimately, the fully simplified version remains:

\[ \frac{3(x + 6)}{4(x - 6)(x^2 + 6x + 36)} \]

Please Verify Again

If there are specific options provided that might yield directly as an answer, please share them as we can further refine or check appropriately. I’m here to help with ensuring this meets your needs correctly! Thank you for your patience!