Answer:
Compare LHS and RHS to prove the statement.
Step-by-step explanation:
Given: a + b + c = 0
We have to show that ![$ \frac{1}{1 + x^b + x^{-c}} + \frac{1}{1 + x^c + x^{-a}} + \frac{1}{1 + x^a + x^{-b}} = 1$](https://tex.z-dn.net/?f=%24%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5Eb%20%2B%20x%5E%7B-c%7D%7D%20%2B%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5Ec%20%2B%20x%5E%7B-a%7D%7D%20%2B%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5Ea%20%2B%20x%5E%7B-b%7D%7D%20%3D%201%24)
We take LCM, simplify the terms and compare LHS and RHS. We will see that LHS = RHS and the statement will be proved.
Taking LCM, we get:
= 1
⇒ ![(1 + x^c + x^{-a})(1 + x^a + x^{-b}) + (1 + x^b + x^{-c})(x^a + x^{-b} + 1) + (x^b + x^{-c} + 1)(x^c + x^{-a} + 1) = (1 + x^a + x^{-b})(1 + x^b + x^{-c})(1 + x^c + x^{-a})](https://tex.z-dn.net/?f=%281%20%2B%20x%5Ec%20%2B%20x%5E%7B-a%7D%29%281%20%2B%20x%5Ea%20%2B%20x%5E%7B-b%7D%29%20%2B%20%281%20%2B%20x%5Eb%20%2B%20x%5E%7B-c%7D%29%28x%5Ea%20%2B%20x%5E%7B-b%7D%20%2B%201%29%20%2B%20%28x%5Eb%20%2B%20x%5E%7B-c%7D%20%2B%201%29%28x%5Ec%20%2B%20x%5E%7B-a%7D%20%2B%201%29%20%20%20%3D%20%281%20%2B%20x%5Ea%20%2B%20x%5E%7B-b%7D%29%281%20%2B%20x%5Eb%20%2B%20x%5E%7B-c%7D%29%281%20%2B%20x%5Ec%20%2B%20x%5E%7B-a%7D%29)
We simplify each term and then compare LHS and RHS.
Simplifying the first term:
![(1 + x^c + x^{-a})(1 + x^a + x^{-b})](https://tex.z-dn.net/?f=%281%20%2B%20x%5Ec%20%2B%20x%5E%7B-a%7D%29%281%20%2B%20x%5Ea%20%2B%20x%5E%7B-b%7D%29)
= ![$ x^{c + a} + x^{c - b} + x^c + 1 + x^{-a - b} + x^{-a} + x^{a} + x^{-b} + 1 $](https://tex.z-dn.net/?f=%24%20x%5E%7Bc%20%2B%20a%7D%20%2B%20x%5E%7Bc%20-%20b%7D%20%2B%20x%5Ec%20%2B%20%201%20%2B%20x%5E%7B-a%20-%20b%7D%20%2B%20x%5E%7B-a%7D%20%2B%20x%5E%7Ba%7D%20%2B%20x%5E%7B-b%7D%20%2B%201%20%24)
= ![$ x^{c + a} + x^{c - b} + x^{c} + x^{-a - b} + x^{-a} + x^{a} + x^{-b} + 2 \hspace{5mm} \hdots (A)$](https://tex.z-dn.net/?f=%24%20x%5E%7Bc%20%2B%20a%7D%20%2B%20x%5E%7Bc%20-%20b%7D%20%2B%20x%5E%7Bc%7D%20%2B%20x%5E%7B-a%20-%20b%7D%20%2B%20x%5E%7B-a%7D%20%2B%20x%5E%7Ba%7D%20%2B%20x%5E%7B-b%7D%20%2B%202%20%20%5Chspace%7B5mm%7D%20%5Chdots%20%28A%29%24)
Now, we simplify the second term we have:
![$ (1 + x^b + x^{-c})(x^a + x^{-b} + 1) $](https://tex.z-dn.net/?f=%24%20%281%20%2B%20x%5Eb%20%2B%20x%5E%7B-c%7D%29%28x%5Ea%20%2B%20x%5E%7B-b%7D%20%2B%201%29%20%20%24)
= ![$ x^{a + b} + 1 + x^{b} + x^{a - c} + x^{-b - c} + x^{-c} + x^{a} + x^{-b} + 1 $](https://tex.z-dn.net/?f=%24%20x%5E%7Ba%20%2B%20b%7D%20%2B%201%20%2B%20x%5E%7Bb%7D%20%2B%20x%5E%7Ba%20-%20c%7D%20%2B%20x%5E%7B-b%20-%20c%7D%20%2B%20x%5E%7B-c%7D%20%2B%20x%5E%7Ba%7D%20%2B%20x%5E%7B-b%7D%20%2B%201%20%24)
= ![$ x^{a + b} + x^{b} + x^{a - c} + x^{- c - b} + x^{-c} + x^a + x^{-b} + 2 \hspace{5mm} \hdots (B) $](https://tex.z-dn.net/?f=%24%20x%5E%7Ba%20%2B%20b%7D%20%2B%20x%5E%7Bb%7D%20%2B%20x%5E%7Ba%20-%20c%7D%20%2B%20x%5E%7B-%20c%20-%20b%7D%20%2B%20x%5E%7B-c%7D%20%2B%20x%5Ea%20%2B%20x%5E%7B-b%7D%20%2B%202%20%5Chspace%7B5mm%7D%20%5Chdots%20%28B%29%20%24)
Again, simplifying
,
= ![$x^{b + c} + x^{b - a} + x^b + 1 + x^{-c -a} + x^{-c} + x^c + x^{-a} + 1 $](https://tex.z-dn.net/?f=%24x%5E%7Bb%20%2B%20c%7D%20%2B%20x%5E%7Bb%20-%20a%7D%20%2B%20x%5Eb%20%2B%201%20%2B%20x%5E%7B-c%20-a%7D%20%2B%20x%5E%7B-c%7D%20%2B%20x%5Ec%20%2B%20x%5E%7B-a%7D%20%2B%201%20%24)
=
(C)
Therefore, LHS = A + B + C
= ![$ x^{c + a} + x^{c - b} + 2x^c + x^{-a - b} + 2x^{-a} + 2x^{a} + 2x^{-b} + x^{a + b} + 2x^b + x^{a - c} + x^{-c -b} + x^{b + c} + x^{b - a} + x^{-c -a} + 6 $](https://tex.z-dn.net/?f=%24%20x%5E%7Bc%20%2B%20a%7D%20%2B%20x%5E%7Bc%20-%20b%7D%20%2B%202x%5Ec%20%2B%20x%5E%7B-a%20-%20b%7D%20%2B%202x%5E%7B-a%7D%20%2B%202x%5E%7Ba%7D%20%2B%202x%5E%7B-b%7D%20%2B%20x%5E%7Ba%20%2B%20b%7D%20%2B%202x%5Eb%20%2B%20x%5E%7Ba%20-%20c%7D%20%2B%20x%5E%7B-c%20-b%7D%20%2B%20x%5E%7Bb%20%2B%20c%7D%20%2B%20x%5E%7Bb%20-%20a%7D%20%2B%20x%5E%7B-c%20-a%7D%20%2B%206%20%24)
Similarly. RHS
= ![$ (x^{b + c} + x^{b - a} + x^b + 1 + x^{-c -a} + x^{-c} + x^c + x^{-a} + 1)(x^a + x^{-b} + 1) $](https://tex.z-dn.net/?f=%24%20%28x%5E%7Bb%20%2B%20c%7D%20%2B%20x%5E%7Bb%20-%20a%7D%20%2B%20x%5Eb%20%2B%201%20%2B%20x%5E%7B-c%20-a%7D%20%2B%20x%5E%7B-c%7D%20%2B%20x%5Ec%20%2B%20x%5E%7B-a%7D%20%2B%201%29%28x%5Ea%20%2B%20x%5E%7B-b%7D%20%2B%201%29%20%24)
Note that if a + b + c = 0, ![$ \implies x^{a + b + c} = x^0 = 1 $](https://tex.z-dn.net/?f=%24%20%5Cimplies%20x%5E%7Ba%20%2B%20b%20%2B%20c%7D%20%3D%20x%5E0%20%3D%201%20%24)
So, RHS = ![x^{c + a} + x^{c - b} + 2x^c + x^{-a - b} + 2x^{-a} + 2x^{a} + 2x^{-b} + x^{a + b} + 2x^b + x^{a - c} + x^{-c -b} + x^{b + c} + x^{b - a} + x^{-c -a} + 6](https://tex.z-dn.net/?f=x%5E%7Bc%20%2B%20a%7D%20%2B%20x%5E%7Bc%20-%20b%7D%20%2B%202x%5Ec%20%2B%20x%5E%7B-a%20-%20b%7D%20%2B%202x%5E%7B-a%7D%20%2B%202x%5E%7Ba%7D%20%2B%202x%5E%7B-b%7D%20%2B%20x%5E%7Ba%20%2B%20b%7D%20%2B%202x%5Eb%20%2B%20x%5E%7Ba%20-%20c%7D%20%2B%20x%5E%7B-c%20-b%7D%20%2B%20x%5E%7Bb%20%2B%20c%7D%20%2B%20x%5E%7Bb%20-%20a%7D%20%2B%20x%5E%7B-c%20-a%7D%20%2B%206)
We see that LHS = RHS.
Therefore, the statement is proved.
Answer:
d+5 or 5+d
Step-by-step explanation:
Sum means to add so if we add the two we get d+5
Answer:
86 meters
Step-by-step explanation:
27+27+16+16=54+32=86 meters
Answer:
-1w=1r
If there are 5 fewer weeds, there will be 5 more roses
Step-by-step explanation:
The question states that the roses increase when the weeds decrease.
Answer: (27, 9)
Multiply the original coordinates by the scale factor 3 to get this answer
9*3 = 27
3*3 = 9
This only works if the center of dilation is the origin.
The general rule is
where k is the scale factor.