Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference. (2 points)

Question

9x^2
+
x + 13

– 2x^2
+ 12x
+

x^2
+ 5x
+ 2

Answer 1

To fill in the blanks, we start with the given polynomial expressions and perform the subtraction step-by-step.

The first expression is: \[ 9x^2 + x + 13 \]

The second expression is: \[ 2x^2 + 12x + \text{(blank)} \]

The third expression is: \[ x^2 + 5x + 2 \]

We need to subtract the sum of the second and third expressions from the first expression.

Let's first find the sum of the second and third expressions:

\[ (2x^2 + 12x + \text{(blank)}) + (x^2 + 5x + 2) \]

Which simplifies to: \[ (2x^2 + x^2) + (12x + 5x) + (\text{(blank)} + 2) = 3x^2 + 17x + (\text{(blank)} + 2) \]

Next, we need to perform the subtraction: \[ (9x^2 + x + 13) - (3x^2 + 17x + (\text{(blank)} + 2)) \]

Now, let's distribute the negative sign: \[ 9x^2 + x + 13 - 3x^2 - 17x - (\text{(blank)} + 2) \]

Combining like terms gives: \[ (9x^2 - 3x^2) + (x - 17x) + (13 - (\text{(blank)} + 2)) = 6x^2 - 16x + (13 - \text{(blank)} - 2) \]

Simplifying further, we have: \[ = 6x^2 - 16x + (11 - \text{(blank)}) \]

From the equation provided, we can see that we need to identify the term that would balance this subtraction. Thus: \[ \text{(blank)} = 11 \]

Accordingly, the filled polynomial equation is: \[ 9x^2 + x + 13 - (2x^2 + 12x + 11) - (x^2 + 5x + 2) \]

So if we refill the blanks, we have: \[ 9x^2 + x + 13 - 2x^2 + 12x + 11 \]

Therefore: \[ \text{blank} = 11 \]

And, thus, we finish with: Resulting difference: \[ 6x^2 - 16x + 9 \] (where the constant term comes from taking 11 from 11, summing these up as previously described).

Answer 2

bot the equations need to be separate from each other and be solved separately

Answer 3

Apologies for the confusion earlier. Let's break it down step-by-step and solve each polynomial separately.

Given the following two expressions:

1. First Expression: \[ 9x^2 + x + 13 \]

2. Second Expression: \[ 2x^2 + 12x + \text{(blank)} \]

3. Third Expression: \[ x^2 + 5x + 2 \]

We need to find the result of subtracting the second and third expressions from the first.

Step 1: Find the combined second and third expressions

To do this, we add the second and third expressions:

\[ (2x^2 + 12x + \text{(blank)}) + (x^2 + 5x + 2) \]

This simplifies to:

\[ (2x^2 + x^2) + (12x + 5x) + (\text{(blank)} + 2) = 3x^2 + 17x + (\text{(blank)} + 2) \]

Step 2: Subtract the combined expression from the first expression

Now we can subtract this entire expression from the first expression:

\[ (9x^2 + x + 13) - (3x^2 + 17x + (\text{(blank)} + 2)) \]

Step 3: Distributing the negative sign

Distributing the negative sign gives us:

\[ 9x^2 + x + 13 - 3x^2 - 17x - (\text{(blank)} + 2) \]

Step 4: Combine like terms

Now we combine the like terms:

1. For the \(x^2\) terms: \[ 9x^2 - 3x^2 = 6x^2 \]

2. For the \(x\) terms: \[ x - 17x = -16x \]

3. For the constant terms: \[ 13 - (\text{(blank)} + 2) = 11 - \text{(blank)} \]

Final Result

Putting all of this together, we have:

\[ 6x^2 - 16x + (11 - \text{(blank)}) \]

Step 5: Determine the value for (blank)

To complete the polynomial equation, we need the total to align correctly. For simplicity and based on common constant values:

If we assume that we would like this polynomial to simplify to a common result like \(0\) or another target value.

Let’s assume we want it zeroed out for a balanced equation, we would set:

\[ 11 - \text{(blank)} = 0 \implies \text{(blank)} = 11 \]

Conclusion

Thus, the value of \(\text{(blank)}\) is \(11\).

Now, we can rewrite the equation and show each of the pieces separately as follows:

1. First Expression: \[ 9x^2 + x + 13 \]

2. Second Expression: \[ 2x^2 + 12x + 11 \]

3. Third Expression: \[ x^2 + 5x + 2 \]

And when we calculate: \[ (9x^2 + x + 13) - (2x^2 + 12x + 11 + x^2 + 5x + 2) \Rightarrow 6x^2 - 16x + 0 \]

This simplifies to: \[ 6x^2 - 16x \]

If you need to separate the steps further or clarify anything else, please let me know!