2/3 of a cubic yard = 6'x6'x4" 1 cubic yard = 9'x9'x4" 1/3 cubic yard = 3'x3'x4"
x8-1 becomes (x4 + 1) (x4 - 1) - perfect square and then (x4 + 1) (x2 - 1)(x2 + 1) - perfect square again. and then (x4 + 1)(x2 + 1)(x + 1)(x - 1) - perfect square again
6 x2 12 x4 48 x2 96 x4 384
6
x4 + 3x3 - x2 - 9x - 6 = 0 x4 + x3 + 2x3 + 2x2 - 3x2 - 3x - 6x - 6 = 0 x3(x + 1) + 2x2(x + 1) - 3x(x + 1) - 6(x + 1) = 0 (x + 1)(x3 + 2x2 - 3x - 6) = 0 (x + 1)[x2(x + 2) - 3(x + 2)] = 0 (x + 1)(x + 2)(x2 - 3) = 0 So x + 1 = 0 so that x = -1 or x + 2 = 0 so that x = -2 or x2 - 3 = 0 so that x = +/- sqrt(3)
2/3 of a cubic yard = 6'x6'x4" 1 cubic yard = 9'x9'x4" 1/3 cubic yard = 3'x3'x4"
6''x4'' and 6 1/2''
0.3333
1/(5x4) 1/20 (1/5)X4 4/5
1
x8-1 becomes (x4 + 1) (x4 - 1) - perfect square and then (x4 + 1) (x2 - 1)(x2 + 1) - perfect square again. and then (x4 + 1)(x2 + 1)(x + 1)(x - 1) - perfect square again
x4 + x2 - 42 Let x2 = t, so that x4 = t2 t2 + t - 42 since -42 = 7(-6) and 7 + (-6) = 1, then t2 + t - 42 = (t - 6)(t + 7) = (x2 - 6)(x2 + 7) By replacing t with x2. So we have, x4 + x2 - 42 = (x2 - 6)(x2 + 7) = [x2 - (square root of 6)2](x2 + 7) = (x - sq. root of 6 )(x + sq. root of 6)(x2 + 7)
Limx→0 [ 1 / (x - 4) + 1 / (x + 4) ] / x = Limx→0 1 / (x2 - 4x) + 1 / (x2 + 4x) = Limx→0 (x2 + 4x) / (x4 - 16x2) + (x2 - 4x) / (x4 - 16x2) = Limx→0 (x2 + 4x - 4x + x2) / (x4 - 16x2) = Limx→0 2x2 / (x4 - 16x2) = Limx→0 2 / (x2 - 16) = 2 / (0 - 16) = -1/8
The infinite series is 1 - x2/2! + x4/4! - x6/6! + ...
6 x2 12 x4 48 x2 96 x4 384
1 - x4 = (1 - x2)(1 + x2) = (1 - x)(1 + x)(1 + x2) (difference of squares)
(3x - 3)(x/6)(x2 - x)= (3)(x - 1)(x/6)(x)(x - 1) = (1/2)(x2)(x -1)2 = (1/2)(x2)(x2 - 2x + 1) = (1/2)x4 - x3 + (1/2)x2