Which of the following expressions are equivalent to 48a^3 - 75a? Select all that apply
A)3(48a^3-75a)
B)3a(16a^2-25)
C)3a(4a+5)(4a+5)
D)3a(4a+5)(4a-5)
E)-3a(25-16a^2)
F)-3a(5-4a)(5+4a)
Show your work and choose all that apply and explain why each answer is incorrect or correct
To determine which expressions are equivalent to 48a^3 - 75a, we can first factor out the greatest common factor, which in this case is 3a:
48a^3 - 75a
= 3a(16a^2 - 25)
Now let's analyze each given expression:
A) 3(48a^3 - 75a)
This expression is not equivalent to 48a^3 - 75a because the 3 is outside the parenthesis and does not affect the terms inside.
B) 3a(16a^2 - 25)
This is the correct factored form derived from the original expression. The expression is equivalent to 48a^3 - 75a.
C) 3a(4a+5)(4a+5)
This expression is incorrect because when expanded, it results in (12a^2 + 15a)(4a + 5), which does not simplify to the original expression.
D) 3a(4a+5)(4a-5)
This expression is incorrect because when expanded, it results in (12a^2 + 15a)(16a - 20a), which does not simplify to the original expression.
E) -3a(25 - 16a^2)
This expression is incorrect because when expanded, it results in (-75a + 48a^3), which is not the original expression.
F) -3a(5-4a)(5+4a)
This expression is also correct and equivalent to 48a^3 - 75a. The negative sign does not affect the final result.
Therefore, the correct equivalent expressions are:
B) 3a(16a^2 - 25)
F) -3a(5-4a)(5+4a)