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To add the polynomial expressions \(2r^2 - 3r + 4\) and \(-2r^2 + 3r + 6\), you combine like terms:
\[ (2r^2 + (-2r^2)) + (-3r + 3r) + (4 + 6) \]
Now, let’s simplify each part:
Combining all these results:
\[ 0 + 0 + 10 = 10 \]
Thus, the result of adding the two polynomials is:
\[ \boxed{10} \]
To add the expressions \(-3k + 4k^3 - 2\) and \(9k^2 - 3k^3 + 4k - 3\), first, we will rewrite them and group like terms.
The first expression is: \[ 4k^3 + 9k^2 - 3k - 2 \]
The second expression is: \[ -3k^3 + 9k^2 + 4k - 3 \]
Now, we will combine them:
\[ (4k^3 + (-3k^3)) + (0 + 9k^2) + (-3k + 4k) + (-2 + (-3)) \]
Now, let's add the coefficients of the like terms:
Now, combine all the results:
\[ k^3 + 9k^2 + k - 5 \]
Thus, the result of adding the two expressions is:
\[ \boxed{k^3 + 9k^2 + k - 5} \]
If you would like the final polynomial expression to be written in a way that emphasizes positive terms, we can rearrange and modify the expression slightly. Remember, polynomials don't inherently have "positive" or "negative" forms; however, we can write the terms in a way that presents them more positively.
The expression we found was:
\[ k^3 + 9k^2 + k - 5 \]
To maintain positivity in presentation, we can factor out \(-1\) from the negative term or rewrite it. However, note that this does not change the reality of the polynomial itself—it's still the same polynomial.
Thus, we can write it as:
\[ -(5 - k^3 - 9k^2 - k) \]
If you'd prefer to present it emphasizing the positive aspects, instead of that step, we might simply note how the expression can be evaluated based on positive \(k\).
However, the standard representation of the polynomial in its simplest form is more conventional as:
\[ k^3 + 9k^2 + k - 5 \]
This polynomial has a mix of positive and negative coefficients. If you want all the terms to be summed positively without altering the polynomial's value, we might consider alternative frames (like evaluating for specific values), but mathematically, the polynomial remains as previously stated.
If you have any additional instructions or specific formats you're looking for, please let me know!
To simplify the expression \((9x^2 - 6x + 2) - (x^2 + 4x - 39)\), we first distribute the negative sign across the second polynomial:
\[ 9x^2 - 6x + 2 - x^2 - 4x + 39 \]
Now, combine like terms:
Putting it all together, we get:
\[ 8x^2 - 10x + 41 \]
So the simplified expression is:
\[ \boxed{8x^2 - 10x + 41} \]
To simplify the expression \((\frac{1}{4}x^2 - 3x + 2.25) + (0.75x^2 + 2x - \frac{1}{4})\), we'll first convert all terms to have the same format and combine like terms.
Now the expression is:
\[ \left(\frac{1}{4}x^2 - 3x + 2.25\right) + \left(\frac{3}{4}x^2 + 2x - \frac{1}{4}\right) \]
For the \(x^2\) terms: \[ \frac{1}{4}x^2 + \frac{3}{4}x^2 = \frac{4}{4}x^2 = 1x^2 \text{ or simply } x^2 \]
For the \(x\) terms: \[ -3x + 2x = -x \]
For the constant terms: Convert \(2.25\) to a fraction: \[ 2.25 = \frac{9}{4} \] Now, combine the constant terms: \[ \frac{9}{4} - \frac{1}{4} = \frac{9 - 1}{4} = \frac{8}{4} = 2 \]
Putting it all together, we have:
\[ x^2 - x + 2 \]
Thus, the simplified expression is:
\[ \boxed{x^2 - x + 2} \]
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