Absolutely, let's determine which of the given expressions are differences of squares.
A difference of squares is defined by the formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, we need to check if each of the given expressions can be written in this form.
1. Expression: [tex]\( p^2 - 40 \)[/tex]
Factor form: [tex]\( p^2 - (2\sqrt{10})^2 \)[/tex]
- [tex]\( p^2 \)[/tex] is a perfect square, yet [tex]\( 40 \)[/tex] is not a perfect square.
Conclusion: This expression cannot be easily factored into the form [tex]\( (a - b)(a + b) \)[/tex] with rational numbers. Therefore, it is not a difference of squares.
2. Expression: [tex]\( n^2 - 225 \)[/tex]
Factor form: [tex]\( n^2 - 15^2 \)[/tex]
- [tex]\( n^2 \)[/tex] is a perfect square.
- [tex]\( 225 = 15^2 \)[/tex], and [tex]\( 225 \)[/tex] is also a perfect square.
Conclusion: This expression can be factored as [tex]\( (n - 15)(n + 15) \)[/tex]. Therefore, it is a difference of squares.
3. Expression: [tex]\( k^2 + 81 \)[/tex]
Factor form: [tex]\( k^2 + (9i)^2 \)[/tex]
- [tex]\( k^2 \)[/tex] is a perfect square.
- [tex]\( 81 \)[/tex] is a perfect square, yet the addition indicates it is not a difference (i.e., [tex]\(a^2 - b^2\)[/tex]).
Conclusion: Since we are dealing with addition, this is clearly not a difference of squares.
4. Expression: [tex]\( t^2 - 1 \)[/tex]
Factor form: [tex]\( t^2 - 1^2 \)[/tex]
- [tex]\( t^2 \)[/tex] is a perfect square.
- [tex]\( 1 = 1^2 \)[/tex], and [tex]\( 1 \)[/tex] is also a perfect square.
Conclusion: This expression can be factored as [tex]\( (t - 1)(t + 1) \)[/tex]. Therefore, it is a difference of squares.
In summary, the expressions that are differences of squares are:
- [tex]\( n^2 - 225 \)[/tex]
- [tex]\( t^2 - 1 \)[/tex]
