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2 years ago

Question 15(Multiple Choice Worth 5 points)

Mathematics

1 answer:

Softa [21]2 years ago

5 0

Equation 2 cuz I’m big brain

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Help me out guys. Got stuck over this algebra question

vovikov84 [41]

Answer:

Compare LHS and RHS to prove the statement.

Step-by-step explanation:

Given: a + b + c = 0

We have to show that $\frac{1}{1 + x^b + x^{-c}} + \frac{1}{1 + x^c + x^{-a}} + \frac{1}{1 + x^a + x^{-b}} = 1$

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:

$\frac{(1 + x^c + x^{-a})(1 + x^a + x^{-b}) + (1 + x^b + x^{-c})(x^a + x^{-b} + 1) + (x^b + x^{-c} + 1)(x^c + x^{-a} + 1)}{(1 + x^a + x^{-b})(1 + x^b + x^{-c})(1 + x^c + x^{-a})}$ = 1

⇒ $(1 + x^c + x^{-a})(1 + x^a + x^{-b}) + (1 + x^b + x^{-c})(x^a + x^{-b} + 1) + (x^b + x^{-c} + 1)(x^c + x^{-a} + 1) = (1 + x^a + x^{-b})(1 + x^b + x^{-c})(1 + x^c + x^{-a})$

We simplify each term and then compare LHS and RHS.

Simplifying the first term:

$(1 + x^c + x^{-a})(1 + x^a + x^{-b})$

= $x^{c + a} + x^{c - b} + x^c + 1 + x^{-a - b} + x^{-a} + x^{a} + x^{-b} + 1$

= $x^{c + a} + x^{c - b} + x^{c} + x^{-a - b} + x^{-a} + x^{a} + x^{-b} + 2 \hspace{5mm} \hdots (A)$

Now, we simplify the second term we have:

$(1 + x^b + x^{-c})(x^a + x^{-b} + 1)$

= $x^{a + b} + 1 + x^{b} + x^{a - c} + x^{-b - c} + x^{-c} + x^{a} + x^{-b} + 1$

= $x^{a + b} + x^{b} + x^{a - c} + x^{- c - b} + x^{-c} + x^a + x^{-b} + 2 \hspace{5mm} \hdots (B)$

Again, simplifying $(x^b + x^{-c} + 1)(x^c + x^{-a} + 1)$ ,

= $x^{b + c} + x^{b - a} + x^b + 1 + x^{-c -a} + x^{-c} + x^c + x^{-a} + 1$

= $x^{b + c} + x^{b - a} + x^b + x^{-c -a} + x^{-c} + x^c + x^{-a} + 2$ (C)

Therefore, LHS = A + B + C

= $x^{c + a} + x^{c - b} + 2x^c + x^{-a - b} + 2x^{-a} + 2x^{a} + 2x^{-b} + x^{a + b} + 2x^b + x^{a - c} + x^{-c -b} + x^{b + c} + x^{b - a} + x^{-c -a} + 6$

Similarly. RHS

= $(x^{b + c} + x^{b - a} + x^b + 1 + x^{-c -a} + x^{-c} + x^c + x^{-a} + 1)(x^a + x^{-b} + 1)$

Note that if a + b + c = 0, $\implies x^{a + b + c} = x^0 = 1$

So, RHS = $x^{c + a} + x^{c - b} + 2x^c + x^{-a - b} + 2x^{-a} + 2x^{a} + 2x^{-b} + x^{a + b} + 2x^b + x^{a - c} + x^{-c -b} + x^{b + c} + x^{b - a} + x^{-c -a} + 6$

We see that LHS = RHS.

Therefore, the statement is proved.

3 0

4 years ago

The ratio of damaged to undamaged toys was 3/17. If there were a total of 200 toys, how many

vagabundo [1.1K]

Answer:

What I would do is find how many undamaged toys per 1 toy and then times that by 200.

Step-by-step explanation:

8 0

3 years ago

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The experimental probability of a coin landing on heads is 7/12. If the coin landed on tails 30 times, find the number of tosses

nordsb [41]

Answer: 72 tosses

Explanation: If the probability of landing heads is 7/12, the probability of landing tails is 5/12. So then you do cross multiplication, where you multiply the 12 and the 30, equalling 360. Then divide that by 5, equalling 72.

4 0

4 years ago

Sam put a 12-inch ruler into a cup. If 3 inches of the ruler were sticking out of the top of the cup, how deep was the cup?

Goryan [66]

Answer:

9 inches deep

Step-by-step explanation:

12 inches - 3 inches = 9 inches

8 0

3 years ago

12. Determine the area of the given parallelogram with length 11 and altitude 5.

Sloan [31]

A= base times height
5times 11
55

4 0

3 years ago

Read 2 more answers