Question

question 6

simplify: \\(-\left(2x^2 - 9\
ight)\\)

\\(\bigcirc -2x^2 - 9\\)
\\(\bigcirc -2x^2 + 9\\)
\\(\bigcirc 2x^2 + 9\\)
\\(\bigcirc -7x^2\\)

question 7

add the following polynomials: \\(\left(3x^4 - 2x^2 + 7\
ight) + \left(2x^3 + 8x^2 + x + 11\
ight)\\)

\\(\bigcirc 3x^4 + 2x^3 + 6x^2 + x + 18\\)
\\(\bigcirc 5x^4 + 6x^2 + x + 18\\)
\\(\bigcirc 3x^4 + 2x^3 + 10x^2 + x + 18\\)
\\(\bigcirc 3x^4 + 8x^2 + x + 11\\)

Explanation:

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"Combining Like Terms",
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"Polynomial Simplification"
]
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</pre_analysis>

<reasoning>

Distribute the negative sign

Using the Distributive Property with Negative Sign knowledge point
\[
-(2x^2 - 9) = -1 \cdot (2x^2) - 1 \cdot (-9) = -2x^2 + 9
\]

Group like terms for addition

Using the Polynomial Addition and Combining Like Terms knowledge points
\[

$$\begin{aligned} &(3x^4 - 2x^2 + 7) + (2x^3 + 8x^2 + x + 11) \\ &= 3x^4 + 2x^3 + (-2x^2 + 8x^2) + x + (7 + 11) \end{aligned}$$

\]

Combine the coefficients

Using the Combining Like Terms knowledge point
\[

$$\begin{aligned} &-2x^2 + 8x^2 = 6x^2 \\ &7 + 11 = 18 \\ &\text{Result: } 3x^4 + 2x^3 + 6x^2 + x + 18 \end{aligned}$$

\]
</reasoning>

<answer>

Question 6

<mcq-option>(A) \(-2x^2 - 9\)</mcq-option>
<mcq-correct>(B) \(-2x^2 + 9\)</mcq-correct>
<mcq-option>(C) \(2x^2 + 9\)</mcq-option>
<mcq-option>(D) \(-7x^2\)</mcq-option>

Question 7

<mcq-correct>(A) \(3x^4 + 2x^3 + 6x^2 + x + 18\)</mcq-correct>
<mcq-option>(B) \(5x^4 + 6x^2 + x + 18\)</mcq-option>
<mcq-option>(C) \(3x^4 + 2x^3 + 10x^2 + x + 18\)</mcq-option>
<mcq-option>(D) \(3x^4 + 8x^2 + x + 11\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Addition"
]
}
</post_analysis>

Answer:

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"needs_drawing": false,
"concepts_used": [
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"Combining Like Terms",
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"Polynomial Simplification"
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}
</pre_analysis>

<reasoning>

Distribute the negative sign

Using the Distributive Property with Negative Sign knowledge point
\[
-(2x^2 - 9) = -1 \cdot (2x^2) - 1 \cdot (-9) = -2x^2 + 9
\]

Group like terms for addition

Using the Polynomial Addition and Combining Like Terms knowledge points
\[

$$\begin{aligned} &(3x^4 - 2x^2 + 7) + (2x^3 + 8x^2 + x + 11) \\ &= 3x^4 + 2x^3 + (-2x^2 + 8x^2) + x + (7 + 11) \end{aligned}$$

\]

Combine the coefficients

Using the Combining Like Terms knowledge point
\[

$$\begin{aligned} &-2x^2 + 8x^2 = 6x^2 \\ &7 + 11 = 18 \\ &\text{Result: } 3x^4 + 2x^3 + 6x^2 + x + 18 \end{aligned}$$

\]
</reasoning>

<answer>

Question 6

<mcq-option>(A) \(-2x^2 - 9\)</mcq-option>
<mcq-correct>(B) \(-2x^2 + 9\)</mcq-correct>
<mcq-option>(C) \(2x^2 + 9\)</mcq-option>
<mcq-option>(D) \(-7x^2\)</mcq-option>

Question 7

<mcq-correct>(A) \(3x^4 + 2x^3 + 6x^2 + x + 18\)</mcq-correct>
<mcq-option>(B) \(5x^4 + 6x^2 + x + 18\)</mcq-option>
<mcq-option>(C) \(3x^4 + 2x^3 + 10x^2 + x + 18\)</mcq-option>
<mcq-option>(D) \(3x^4 + 8x^2 + x + 11\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Addition"
]
}
</post_analysis>