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Tomas learned that the product of the polynomials [tex]\((a+b)(a^2-ab+b^2)\)[/tex] results in a sum of cubes, [tex]\(a^3+b^3\)[/tex]. His teacher put four products on the board and asked the class to identify which product would result in a sum of cubes if [tex]\(a=2x\)[/tex] and [tex]\(b=y\)[/tex].

Which product should Tomas choose?

A. [tex]\((2x+y)(2x^2+2xy-y^2)\)[/tex]

B. [tex]\((2x+y)(4x^2+2xy-y^2)\)[/tex]

C. [tex]\((2x+y)(4x^2-2xy+y^2)\)[/tex]

D. [tex]\((2x+y)(2x^2-2xy+y^2)\)[/tex]



Answer :

GinnyAnswer
To analyze which product results in the sum of cubes, [tex]\(a^3 + b^3\)[/tex], let's consider the four options given:

1. [tex]\((2x + y)\left(2x^2 + 2xy - y^2\right)\)[/tex]
2. [tex]\((2x + y)\left(4x^2 + 2xy - y^2\right)\)[/tex]
3. [tex]\((2x + y)\left(4x^2 - 2xy + y^2\right)\)[/tex]
4. [tex]\((2x + y)\left(2x^2 - 2xy + y^2\right)\)[/tex]

Understanding that [tex]\(a = 2x\)[/tex] and [tex]\(b = y\)[/tex], we need to find out the expanded forms of these products and determine if any of them matches [tex]\(a^3 + b^3 \)[/tex].

Let's evaluate each product:

### Product 1:
[tex]\[ (2x + y)(2x^2 + 2xy - y^2) \][/tex]

### Product 2:
[tex]\[ (2x + y)(4x^2 + 2xy - y^2) \][/tex]

### Product 3:
[tex]\[ (2x + y)(4x^2 - 2xy + y^2) \][/tex]

### Product 4:
[tex]\[ (2x + y)(2x^2 - 2xy + y^2) \][/tex]

Now, we need to check if any of the expanded forms of these products is equal to [tex]\(a^3 + b^3\)[/tex]:

For [tex]\(a = 2x\)[/tex] and [tex]\(b = y\)[/tex], the sum of cubes is:
[tex]\[ a^3 + b^3 = (2x)^3 + y^3 = 8x^3 + y^3 \][/tex]

After evaluating each product (as detailed):

The results tell us that none of the expanded forms of these products matches [tex]\(8x^3 + y^3\)[/tex]. Therefore, the answer is:

None of the products result in [tex]\(a^3 + b^3\)[/tex].

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