Which products result in a difference of squares? check all that apply. (5z 3)(–5z – 3) (w – 2.5)(w 2.5) (8g 1)(8g 1) (–4v – 9)(–4v 9) (6y 7)(7y – 6) (p – 5)(p – 5)
The Correct Answer and Explanation is :
The difference of squares occurs when two binomials are multiplied, and the product is a result of subtracting squares of terms, i.e., ((a – b)(a + b) = a^2 – b^2). Let’s examine each product and check if it results in a difference of squares:
- ((5z + 3)(-5z – 3)):
This expression represents a difference of squares because it fits the form ((a – b)(a + b) = a^2 – b^2). In this case:
- (a = 5z) and (b = 3)
- Using the formula: ((5z)^2 – 3^2 = 25z^2 – 9)
Therefore, this product is a difference of squares.
- ((w – 2.5)(w + 2.5)):
This is another difference of squares, as it has the form ((a – b)(a + b)):
- (a = w) and (b = 2.5)
- Applying the formula: (w^2 – (2.5)^2 = w^2 – 6.25)
So, this product is a difference of squares.
- ((8g + 1)(8g + 1)):
This expression is not a difference of squares. It represents the square of a binomial, i.e., ((a + b)(a + b) = a^2 + 2ab + b^2), not the difference between squares. - ((-4v – 9)(-4v + 9)):
This product is a difference of squares, as it follows the form ((a – b)(a + b)):
- (a = -4v) and (b = 9)
- Applying the difference of squares formula: ((-4v)^2 – 9^2 = 16v^2 – 81)
Hence, this is a difference of squares.
- ((6y + 7)(7y – 6)):
This is not a difference of squares. The binomials do not follow the form ((a – b)(a + b)) because both terms involve different variables and constants. - ((p – 5)(p – 5)):
This expression is also not a difference of squares. It represents the square of the binomial ((p – 5)^2 = p^2 – 10p + 25).
Conclusion:
The products that result in a difference of squares are:
- ((5z + 3)(-5z – 3))
- ((w – 2.5)(w + 2.5))
- ((-4v – 9)(-4v + 9))