Long Division:
5X^3 - 3X^2 - 4
---------------------
X^2
Commutative property: you can split this into two fractions:
5x
3 - 3x
2 4
---------- - -----
x
2 x
2Factor out x
2, leaving you with :
5x - 3 - 4/x
2That's the most-simplified as that can get, and arguably the first form is even better.
Solve:
5x-4 = 2/3
Multiply both sides by 3:
15x - 12 = 2
Add 12 to both sides
15x = 14
Divide both sides by 15
x = 14/15
5x+4
1 3 4
___ + __ = __
x-2 x+6 (x+3)(x-2)
I don't know what you want out of those.... The second one, hrm. Multiply both sides by (x+3)(x-2)
1(x+3) + [3(x
2 - 5x - 6) / (x+6)] = 4
Make into common terms by multiplying 1(x+3) and 4 by (x+6)/(x+6)
[(x
2 + 9x + 18)/(x+6)] + [3(x
2 - 5x - 6)/(x+6)] = [(4x + 24)/(x+6)]
Multiply both sides by (x+6) to get a quadratic equation:
x
2 + 9x + 18 + 3x
2 - 5x - 6 = 4x + 24
Simplify
4x
2 + 4x + 12 = 4x + 24
Divide both sides by 4:
x
2 + x + 3 = x + 6
Subtract (x+6) from both sides:
x
2 - 3 = 0
Solution: x
2 = 3, so x = +/- sqrt(3)
Simplify:
1) (9x+81)/(x^2-9) * (2x^2-6x)/(x^2+18x+81)
Separate each polynomial term into factors:
[9(x+9)] / [(x+3)(x-3)] * [2x(x-3)]/(x+9)
2Cross-cancel: one factor of (x+9), one factor of (x-3):
9/(x+3) * 2x/(x+9)
Multiply:
18x/[(x+3)(x+9)]
= 18x/(x
2+12x+27)
2) (2/x^2-7x) + (3/x-7) - (3/x)
2 / [x(x-7)] + 3/(x-7) - 3/x
Get everything in common terms: multiply the 2nd term by x/x and the third term by (x-7)/(x-7)
2/[x(x-7)] + 3x/[x(x-7)] - (3x-21)/[x(x-7)]
Combine like terms:
-19 / [x(x-7)], or -19/(x
2 - 7x)
3) (1/x) + (1/y)
---------------------
(2/x) - (2/y)
Separate them out.
[(1/x) / [(2/x)-(2/y)]] + [(1/y) / [(2/x)-(2/y)]]
Realize that 1/x means you can multiply the denominator by x
1 / (x[(2/x)-(2/y)]) + 1 / (y[(2/x)-(2/y)])
Distribute:
(1 / [2 - 2x/y]) + (1 / [2y/x - 2])
Whether that's actually more simple is anyone's argument