To multiply \( (6x^2 - 7x + 4.7)(x^2 - 2.8) \) using the area model, we need to set it up based on the two polynomials.
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Set up the area model:
We arrange \( (6x^2 - 7x + 4.7) \) vertically and \( (x^2 - 2.8) \) horizontally.
\[ \begin{array}{c|c|c} & x^2 & -2.8 \ \hline 6x^2 & 6x^2 \cdot x^2 & 6x^2 \cdot -2.8 \ \hline -7x & -7x \cdot x^2 & -7x \cdot -2.8 \ \hline 4.7 & 4.7 \cdot x^2 & 4.7 \cdot -2.8 \ \end{array} \]
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Now, fill in each of the response areas:
- Top left: \( 6x^2 \cdot x^2 = 6x^4 \)
- Top middle: \( 6x^2 \cdot -2.8 = -16.8x^2 \)
- Middle left: \( -7x \cdot x^2 = -7x^3 \)
- Middle right: \( -7x \cdot -2.8 = 19.6x \)
- Bottom left: \( 4.7 \cdot x^2 = 4.7x^2 \)
- Bottom middle: \( 4.7 \cdot -2.8 = -13.16 \)
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Complete the area model:
\[ \begin{array}{c|c|c} & x^2 & -2.8 \ \hline 6x^2 & 6x^4 & -16.8x^2 \ \hline -7x & -7x^3 & 19.6x \ \hline 4.7 & 4.7x^2 & -13.16 \ \end{array} \]
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Combine like terms:
Now we need to sum up all the contributions to find the product in standard form:
- From \( x^4 \): \( 6x^4 \)
- From \( x^3 \): \( -7x^3 \)
- From \( x^2 \): \( -16.8x^2 + 4.7x^2 = -12.1x^2 \)
- From \( x \): \( 19.6x \)
- Constant term: \( -13.16 \)
Therefore, the product in standard form is:
\[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]
Final Answers:
- The response areas would be filled with \( 6x^4 \), \( -16.8x^2 \), \( -7x^3 \), \( 19.6x \), \( 4.7x^2 \), \( -13.16 \).
- The product in standard form is \( 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \).
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