Eric multiplied (6x^2−7x+4.7)(x^2−2.8) using an area model.

Question

Complete the blank spaces in the area model and determine the product in standard form.
6x^2 -7x 4.7
x^2 ___ -7x^3 ____
-2.8 ____ ____ -13.16

The product in standard form is

Answer 1

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

1. For the top left rectangle: $$6x^2\cdot x^2=6x^4$$

2. For the top middle rectangle: $$6x^2\cdot (-2.8)=-16.8x^2$$

3. For the middle left rectangle: $$-7x\cdot x^2=-7x^3$$

4. For the middle middle rectangle: $$-7x\cdot (-2.8)=19.6x$$

5. For the bottom left rectangle: $$4.7\cdot x^2=4.7x^2$$

6. 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
---------------------------------------
x^2           | 6x^4      | -7x^3      | 4.7x^2
---------------------------------------
-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: $$6x^4-7x^3-12.1x^2+19.6x-13.16$$