ric multiplied (6x2−7x+4.7)(x2−2.8)

Question

using an area model.

Complete the blank spaces in the area model and determine the product in standard form.

(5 points)
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6x2
−7x
4.7
x2
−7x3
Response area
−2.8
−13.16
The product in standard form is

Answer 1

To multiply the two polynomials \( (6x^2 - 7x + 4.7)(x^2 - 2.8) \) using an area model, we first set up a grid where we break the polynomials into their individual terms and multiply each term accordingly.

1. Set up the area model:

   • On the top, put the terms of the first polynomial: \( 6x^2 \), \( -7x \), and \( 4.7 \).
   • On the side, put the terms of the second polynomial: \( x^2 \) and \( -2.8 \).

2. Now fill in the area grid by multiplying each pair of terms:

• 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 middle: \( -7x \cdot (-2.8) = 19.6x \)
• Bottom left: \( 4.7 \cdot x^2 = 4.7x^2 \)
• Bottom middle: \( 4.7 \cdot (-2.8) = -13.16 \)

Now, let’s fill in the area model:

• \( 6x^4 \) in the top left.
• \( -16.8x^2 \) in the top middle.
• \( -7x^3 \) in the middle left.
• \( 19.6x \) in the middle middle.
• \( 4.7x^2 \) in the bottom left.
• \( -13.16 \) in the bottom middle.

3. Now combine all terms based on their degrees:

• The \( x^4 \) term: \( 6x^4 \)
• The \( x^3 \) term: \( -7x^3 \)
• The \( x^2 \) terms: \( -16.8x^2 + 4.7x^2 = -12.1x^2 \)
• The \( x \) term: \( 19.6x \)
• The constant term: \( -13.16 \)

Combining these, the product of the two polynomials in standard form is: \[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]

So the final response for the product in standard form is: \[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]