Emily and Zach have two different polynomials to multiply: Polynomial product A:
(4x2 – 4x)(x2 – 4) Polynomial product B:
(x2 + x – 2)(4x2 – 8x)
They are trying to determine if the products of the two polynomials are the same. But they disagree about how to solve this problem.

1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)

Classmate Conjecture
Emily

Zach

Evaluating the Conjectures

Who do you think is correct? (1 point)

Analyzing the Data

Table Method:

2. Use the table to find the products of the two polynomials. Write your answer in descending order. (4 points: 2 points for each product)

A) (4x2 – 4x)(x2 – 4)

4x2 –4x
x2
–4

B) (x2 + x – 2)(4x2 – 8x)






3. Are the two products the same when you multiply them with the table method? (1 point)

Multiplying Using the Distributive Property:

4. Multiply the two polynomials using the distributive property. (4 points: 2 points for each product)

A) (4x2 – 4x)(x2 – 4)

B) (x2 + x – 2)(4x2 – 8x)

5. Are the two products the same when you multiply them horizontally? (1 point)

Multiplying Vertically:

6. Find the product for both sets of polynomials below by multiplying vertically. (4 points: 2 points for each product)
A)

B)

7. Are the two products the same when you multiply them vertically? (1 point)

Making a Decision:

8. Who was right, Emily or Zach? Are the products the same with the three different methods of multiplication? (1 point)

9. Which of these three methods is your preferred method for multiplying polynomials? Why? (1 point)

Polynomial product A:

(4x2 – 4x)(x2 – 4) Polynomial product B:

(x2 + x – 2)(4x2 – 8x)

They are trying to determine if the products of the two polynomials are the same. But they disagree about how to solve this problem.

1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)

Classmate Conjecture

Emily: says they should just check one method

Zach: says that they should check all three bc there could be multiple answers

Evaluating the Conjectures

Who do you think is correct? (1 point)

If they do all of the methods correctly, they should get the same answer for all of them.

Analyzing the Data

Table Method:

2. Use the table to find the products of the two polynomials. Write your answer in descending order. (4 points: 2 points for each product)

A) (4x2 – 4x)(x2 – 4)

         4x2        –4x

x2      4x^4     –4x^3

–4     -16x^2     16x

4x^4 - 4x^3 - 16x^2 + 16x

B) (x2 + x – 2)(4x2 – 8x)

              x^2           x            -2

4x^2       4x^4      4x^3       -8x^2

-8x       -8x^3       -8x^2       16x

4x^4 - 4x^3 - 8x^2 - 8x^3 - 8x^2 +16x

combine like terms

4x^4 - 4x^3 - 16x^2 + 16x

3. Are the two products the same when you multiply them with the table method? (1 point)

yes

Multiplying Using the Distributive Property:

4. Multiply the two polynomials using the distributive property. (4 points: 2 points for each product)

5. Are the two products the same when you multiply them horizontally? (1 point)

yes

Making a Decision:

8. Who was right, Emily or Zach? Are the products the same with the three different methods of multiplication? (1 point)

Emily was correct. the products are the same no matter what method.

9. Which of these three methods is your preferred method for multiplying polynomials? Why? (1 point)

personally, i like the distributive property, but do whichever one you like.

hiking down a winding trail.

step-by-step explanation: