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I need to express each difference in simpest form.
This problem is a RAtional expression. To subtract, we have to make sure
that the denominators are the same. If not, we will find the LCD.
d^2+3 d - 4
_________ - _______
d^2 - 2d d
d^2 + 3 d - 4
_________ - _______ LCD is d(d - 2)
d(d - 2) d
Make an equivalent expression using the LCD
d^2 + 3 (d - 4)(d - 2)
________ - _______________
d(d - 2) d(d - 2)
d^2 + 3 d^2 - 6d + 8
________ - _______________ Subtract the numerator, copy the denominator
d(d - 2) d(d - 2)
d^2 + 3 - (d^2 - 6d + 8)
__________________________
d (d - 2)
d^2 + 3 - 1(d^2 - 6d + 8)
__________________________
d (d - 2)
d^2 + 3 - d^2 + 6d - 8)
__________________________ Combine like terms
d (d - 2)
6d - 5
________
d(d -2)
and
d^2 - 11 d + 1
______________ - _________
d^2 - 7d +12 d - 4
d^2 - 11 d + 1
______________ - _________ Factor d^2 - 7d +12
(d - 3)(d - 4) d - 4
d^2 - 11 (d + 1)(d - 3)
______________ - ________________ Find equivalent expression using LCD
(d - 3)(d - 4) (d - 3)(d - 4)
d^2 - 11 (d^2 - 2d - 3)
______________ - ________________
(d - 3)(d - 4) (d - 3)(d - 4)
d^2 - 11 - (d^2 - 2d - 3)
____________________________
(d - 3)(d - 4)
d^2 - 11 - 1(d^2 - 2d - 3)
____________________________
(d - 3)(d - 4)
d^2 - 11 - d^2 + 2d + 3)
____________________________ Combine like terms
(d - 3)(d - 4)
2d - 8
_______________ Factor 2d - 8.
(d - 3)(d - 4)
2(d - 4)
_______________ Divide the common factor.
(d - 3)(d - 4)
2 
_______________
(d - 3)
2
_______
d - 3
(