Question

use the vertical method to multiply \\((4a^3 - 2a + 3a^2 + 1)\\) and \\((3 - 2a + a^2)\\). what would be the value of a?

Explanation:

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<reasoning>

Identify the term corresponding to A

The vertical multiplication aligns terms by their degree. The top polynomial is written in standard form:
\[
4a^3 + 3a^2 - 2a + 1
\]
The second row of partial products is obtained by multiplying the top polynomial by the middle term of the multiplier, which is \(-2a\):
\[
-2a \cdot (4a^3 + 3a^2 - 2a + 1) = -8a^4 - 6a^3 + 4a^2 - 2a
\]

Match with the visual layout

Looking at the second row of partial products in the image:
\[
-8a^4 - 6a^3 + A - 2a
\]
Comparing this term-by-term with the calculated product:
\[
A = 4a^2
\]
</reasoning>

<answer>
\(4a^2\)
</answer>

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Answer:

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<reasoning>

Identify the term corresponding to A

The vertical multiplication aligns terms by their degree. The top polynomial is written in standard form:
\[
4a^3 + 3a^2 - 2a + 1
\]
The second row of partial products is obtained by multiplying the top polynomial by the middle term of the multiplier, which is \(-2a\):
\[
-2a \cdot (4a^3 + 3a^2 - 2a + 1) = -8a^4 - 6a^3 + 4a^2 - 2a
\]

Match with the visual layout

Looking at the second row of partial products in the image:
\[
-8a^4 - 6a^3 + A - 2a
\]
Comparing this term-by-term with the calculated product:
\[
A = 4a^2
\]
</reasoning>

<answer>
\(4a^2\)
</answer>

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