A binary operation * is defined on the set T = {-2,-1,1,2} by p*q = p\(^2\) + 2pq – q\(^2\), where p,q ∊ T.
Copy and complete the table.
| * | -2 | -1 | 1 | 2 |
| -2 | 7 | -8 | ||
| -1 | 2 | -2 | ||
| 1 | -7 | 1 | ||
| 2 | -1 |
Explanation
p*q = p\(^2\) + 2pq - q\(^2\)
when p = -2 and q = -2
p*q = -2\(^2\) + 2(-2)(-2) - (-2)\(^2\)
p*q = 4 + 8 - 4 = 8
when p = -2 and q = -1
p*q = -2\(^2\) + 2(-2)(-1) - (-1)\(^2\)
p*q = 4 + 4 - 1 = 7
when p = -2 and q = 1
p*q = -2\(^2\) + 2(-2)(1) - (1)\(^2\)
p*q = 4 - 4 -1 = -1
when p = -1 and q = -2
p*q = -1\(^2\) + 2(-1)(-2) - (-2)\(^2\)
p*q = 1 + 4 - 4 = 1
when p = -1 and q = -1
p*q = -1\(^2\) + 2(-1)(-1) - (-1)\(^2\)
p*q = 1 + 2 -1 = 2
when p = -1 and q = 2
p*q = -1\(^2\) + 2(-1)(2) - (2)\(^2\)
p*q = 1 - 4 - 4 = -7
when p = 1 and q = -2
p*q = 1\(^2\) + 2(1)(-2) - (-2)\(^2\)
p*q = 1 - 4 - 4 = -7
when p = 1 and q = -1
p*q = 1\(^2\) + 2(1)(-1) - (-1)\(^2\)
p*q = 1 - 2 - 1 = -2
when p = 1 and q = 1
p*q = 1\(^2\) + 2(1)(1) - (1)\(^2\)
p*q = 1 + 2 - 1 = 2
when p = 1 and q = 2
p*q = 1\(^2\) + 2(1)(2) - (2)\(^2\)
p*q = 1 + 4 - 4 = 1
when p = 2 and q = -2
p*q = 2\(^2\) + 2(2)(-2) - (-2)\(^2\)
p*q = 4 - 8 - 4 = -8
when p = 2 and q = -1
p*q = 2\(^2\) + 2(2)(-1) - (-1)\(^2\)
p*q = 4 - 4 - 1 = -1
when p = 2 and q = 1
p*q = 2\(^2\) + 2(2)(1) - (1)\(^2\)
p*q = 4 + 4 - 1 = 7
when p = 2 and q = 2
p*q = 2\(^2\) + 2(2)(2) - (2)\(^2\)
p*q = 4 + 8 - 4 = 8
| * | -2 | -1 | 1 | 2 |
| -2 | 8 | 7 | -1 | -8 |
| -1 | 1 | 2 | -2 | -7 |
| 1 | -7 | -2 | 2 | 1 |
| 2 | -8 | -1 | 7 | 8 |