(a) Which of the following are examples of the expression "difference of two cubes"? (Select all that apply.)
(y2 - 9)3
3 - 4y
64 - y3
(y - 1)3
y3 - 27
27 - 3y
y3
(b) Which of the following are examples of the expression "cube of a difference"? (Select all that apply.)
27 - 3y
(y - 1)3
(y2 - 9)3
y3
y3 - 27
64 - y3
3 - 4y
difference of a and b means a-b
difference of two cubes means a^3 - b^3
That would be 64-y^3 or y^3-27
difference means a-b
cube of difference is (a-b)^3
that would be (y^2-9)^3 or (y-1)^3
Which of the following are examples of faulty reasoning? Select all that apply.
To determine whether each expression is an example of the "difference of two cubes" or "cube of a difference," we need to understand the patterns for each.
(a) The "difference of two cubes" pattern can be written as (a^3 - b^3), where a and b are numbers or variables. The expression should match this pattern.
(y^2 - 9)^3: Not an example of the "difference of two cubes" pattern since the exponent is on the entire expression, not just on (y^2 - 9).
3 - 4y: Not an example of the "difference of two cubes" pattern since it is a linear expression without any cubes.
64 - y^3: An example of the "difference of two cubes" pattern since it matches (a^3 - b^3) with a=4 and b=y.
(y - 1)^3: An example of the "difference of two cubes" pattern since it matches (a^3 - b^3) with a=y and b=1.
y^3 - 27: An example of the "difference of two cubes" pattern since it matches (a^3 - b^3) with a=y and b=3.
27 - 3y: Not an example of the "difference of two cubes" pattern since it is a linear expression without any cubes.
y^3: Not an example of the "difference of two cubes" pattern since it is only a cube without being a difference of two cubes.
Therefore, the examples of the "difference of two cubes" in this list are:
64 - y^3
(y - 1)^3
y^3 - 27
(b) The "cube of a difference" pattern can be written as (a - b)^3, where a and b are numbers or variables. The expression should match this pattern.
27 - 3y: Not an example of the "cube of a difference" pattern since it is not in the form (a - b)^3.
(y - 1)^3: Not an example of the "cube of a difference" pattern since it is in the form (a - b)^3, but without any difference inside.
(y^2 - 9)^3: Not an example of the "cube of a difference" pattern since it is not in the form (a - b)^3.
y^3: Not an example of the "cube of a difference" pattern since it is only a cube without being a difference of two terms.
y^3 - 27: Not an example of the "cube of a difference" pattern since it is not in the form (a - b)^3.
64 - y^3: Not an example of the "cube of a difference" pattern since it is not in the form (a - b)^3.
3 - 4y: Not an example of the "cube of a difference" pattern since it is not in the form (a - b)^3.
Therefore, there are no examples of the "cube of a difference" in this list.