To fill in the area model and calculate the product of \((6x^2 - 7x + 4.7)(x^2 - 2.8)\), we'll compute each area of the rectangles in the model and then combine like terms.
Setting Up the Area Model
The rows of the area model represent the first polynomial \( (6x^2 - 7x + 4.7) \) and the columns represent the second polynomial \( (x^2 - 2.8) \).
Calculate Each Area
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For the top left rectangle: \[ 6x^2 \cdot x^2 = 6x^4 \]
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For the top middle rectangle: \[ 6x^2 \cdot (-2.8) = -16.8x^2 \]
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For the middle left rectangle: \[ -7x \cdot x^2 = -7x^3 \]
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For the middle middle rectangle: \[ -7x \cdot (-2.8) = 19.6x \]
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For the bottom left rectangle: \[ 4.7 \cdot x^2 = 4.7x^2 \]
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For the bottom middle rectangle: \[ 4.7 \cdot (-2.8) = -13.16 \]
Filled Area Model
Now we fill in the computed areas into the area model:
| 6x^2 | -7x | 4.7
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x^2 | 6x^4 | -7x^3 | 4.7x^2
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-2.8 | -16.8x^2 | 19.6x | -13.16
Combine Like Terms
Now we will combine all terms from the area model:
- From \(6x^4\): \(6x^4\)
- From \(-7x^3\): \(-7x^3\)
- From the \(x^2\) terms: \[ -16.8x^2 + 4.7x^2 = -12.1x^2 \]
- From the \(x\) terms: \(19.6x\)
- Constant term: \(-13.16\)
Final Expression in Standard Form
The product in standard form will be:
\[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]
Conclusion
Thus, the completed area model and the product of the polynomials in standard form is: \[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]
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