Problem 1

Reduce the following expression to its simplest form:

$open paren 9 a squared b squared minus 4 b to the fourth power close paren times open paren a squared minus b squared close paren minus open paren 3 a b minus 2 b squared close paren times open paren 3 a times open bracket a squared plus b squared close bracket minus 2 b times open bracket b squared plus 3 a b minus a squared close bracket close paren times b$

Solution

Working on the first term. Noting that
$open paren 9 a squared b squared minus 4 b to the fourth power close paren equals b squared times open paren 9 a squared minus 4 b squared close paren$
and
$9 a squared minus 4 b squared equals open paren 3 a close paren squared minus open paren 2 b close paren squared equals open paren 3 a minus 2 b close paren times open paren 3 a plus 2 b close paren$
Hence, the first term becomes
$b squared times open paren 3 a minus 2 b close paren times open paren 3 a plus 2 b close paren times open paren a squared minus b squared close paren$
Now the second term. Write
$3 a b minus 2 b squared equals b times open paren 3 a minus 2 b close paren$
Multiplying by the extra factor $b$ gives
$b squared times open paren 3 a minus 2 b close paren$
Next, simplify
$T equals 3 a times open paren a squared plus b squared close paren minus 2 b times open paren b squared plus 3 a b minus a squared close paren$
Expand to obtain
$3 a times open paren a squared plus b squared close paren equals 3 a cubed plus 3 a b squared$
and
$negative 2 b times open paren b squared plus 3 a b minus a squared close paren equals negative 2 b cubed minus 6 a b squared plus 2 a squared b$
Combining gives
$T equals 3 a cubed plus 2 a squared b minus 3 a b squared minus 2 b raised to the 3 period power$

Thus the second term becomes

$b squared times open paren 3 a minus 2 b close paren times open paren 3 a cubed plus 2 a squared b minus 3 a b squared minus 2 b cubed close paren$

c) Combine the terms. The expression becomes

$E equals b squared times open paren 3 a minus 2 b close paren times open bracket open paren 3 a plus 2 b close paren times open paren a squared minus b squared close paren minus open paren 3 a cubed plus 2 a squared b minus 3 a b squared minus 2 b cubed close paren close bracket$

Expand

$open paren 3 a plus 2 b close paren times open paren a squared minus b squared close paren equals 3 a times open paren a squared minus b squared close paren plus 2 b times open paren a squared minus b squared close paren equals 3 a cubed minus 3 a b squared plus 2 a squared b minus 2 b raised to the 3 period power$ Subtracting the second group results in

$S equals 3 a cubed minus 3 a b squared plus 2 a squared minus 2 b cubed minus 3 a cubed plus 2 a squared b minus 3 a b squared minus 2 b cubed equals 0 period$

d) Since $S equals 0$ , it follows that

$E equals b squared times open paren 3 a minus 2 b close paren times 0 equals 0 period$

Answer: $0$