To multiply the expressions \((6x^2−7x+4.7)(x^2−2.8)\) using an area model, we can break it down into parts. First, we will identify the components of the expression.
The area model involves creating a box for each term in the first expression multiplied by each term in the second expression. We can outline it as follows:
Breaking Down the Expressions
- First expression: \(6x^2, -7x, 4.7\)
- Second expression: \(x^2, -2.8\)
Area Model Setup
-
Top Row:
- \(6x^2\)
- \(-7x\)
- \(4.7\)
-
Side Column:
- \(x^2\)
- \(-2.8\)
Filling in the Affected Areas
-
Area for \(6x^2 \cdot x^2\):
- \(6x^4\)
-
Area for \(6x^2 \cdot -2.8\):
- \(-16.8x^2\)
-
Area for \(-7x \cdot x^2\):
- \(-7x^3\)
-
Area for \(-7x \cdot -2.8\):
- \(19.6x\)
-
Area for \(4.7 \cdot x^2\):
- \(4.7x^2\)
-
Area for \(4.7 \cdot -2.8\):
- \(-13.16\)
Collecting Like Terms
Now we need to gather the terms:
- \(6x^4\) (from the \(6x^2 \cdot x^2\))
- Combine \(-7x^3\) (from \(-7x \cdot x^2\)) with terms from \(4.7 \cdot x^2 + -16.8x^2\):
- \(-7x^3\)
- \(4.7 - 16.8 = -12.1\) for \(x^2\)
- \(19.6x\) (from \(-7x \cdot -2.8\))
- Finally, combine the constant: \(-13.16\)
The Standard Form of the Product
Arranging these in descending order of degree yields the product in standard form:
\[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]
Final Filling of the Response Areas
Using the areas derived:
- The response areas that needed to be filled would therefore include:
- \(6x^4\) for \(6x^2 \cdot x^2\)
- \(-7x^3\) for \(-7x \cdot x^2\)
- \(-12.1x^2\) for the aggregate \(x^2\) term
- \(19.6x\) for \(+19.6x\)
- \(-13.16\) for the constant
Thus, the final expression in standard form is: \[ \boxed{6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16} \]
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