Question

Amber, Bernie, and Carlos are working on a problem together. Their goal is to correctly apply the difference of cubes formula to factor 2x5−250x2.
Here is their plan:
First, factor the greatest common factor out of the expression.
Next, identify a and b.
Finally, use the difference of cubes method to factor the expression.

Each person followed the steps and arrived at a different answer. Read each student’s work and identify who followed the steps correctly.

Amber's work:
First, factor out 2x2:2x5−250x2=2x2(x3−125). Next, identify a and b:a=x, b=125. Finally, follow the pattern to get 2x2(x−125)(x2+125x+15,625).

Bernie's work:
First, factor out x2:2x5−250x2=x2(2x3−250). Next, identify a and b:a=2x; b=250. Finally, follow the pattern to get x2(2x−250)(4x2+500x−62,500).

Carlos' work:
First, factor out 2x2:2x5−250x2=2x2(x3−125). Next, identify a and b:a=x; b=5. Finally, follow the pattern to get 2x2(x−5)(x2+5x+25).

Which statements accurately analyze why each student is correct or incorrect?

There may be more than one correct answer. Select all correct answers.

Amber is incorrect because 125 is equal to b3, not b.
Bernie is incorrect because x2 is not the GCF of the polynomial.
Bernie is correct because he correctly factored the GCF, identified a and b, and applied the difference of cubes method.
Carlos is incorrect because 5 is equal to b√3, not b.
Carlos is correct because he correctly factored the GCF, identified a and b, and applied the difference of cubes method.
Amber is correct because she correctly factored the GCF, identified a and b, and applied the difference of cubes method.

Answer 1

Answer: Carlos is correct

Step-by-step explanation:

2x^5-250x^2

The first step is to factorize out 2x^2

2x^2(x^3-125)

This can be rewritten as

2x^2(x^3-5^3),

Recall, difference if cube (a^3-b^3)

= (a-b)(a^2 +ab + b^2)

from our equation,

a= x

b = 5.

So the difference of cubes will be

2x^2[(x-5) (x^2 + 5x + 5^2)]

=2x^2[(x-5) (x^2 + 5x + 25)]

Carlos followed these steps. So he is correct. Carlos is correct because he correctly factored the GCF, identified a and b, and applied the difference of cubes method.

Amber is incorrect because 125 is equal to b3, not b.

Bernie is incorrect because x2 is not the GCF of the polynomial. The GCF is 2x^2.