We will set a variable, d, to represent the day of the week that January starts on. For instance, if it started on Monday, d + 1 would be Tuesday, d + 2 would be Wednesday, etc. up to d + 6 to represent the last day of the week (in our example, Sunday). The next week would start over at d, and the month would continue. For non-leap years:
If January starts on <u>d</u>, February will start 31 days later. Following our pattern above, this will put it at <u>d</u><u> + 3</u> (28 days would be back at d; 29 would be d+1, 30 would be d+2, and 31 is at d+3). In a non-leap year, February has 28 days, so March will start at <u>d</u><u>+3</u> also. April will start 31 days after that, so that puts us at d+3+3=<u>d</u><u>+6</u>. May starts 30 days after that, so d+6+2=d+8. However, since we only have 7 days in the week, this is actually back to <u>d</u><u>+1</u>. June starts 31 days after that, so d+1+3=<u>d</u><u>+4</u>. July starts 30 days after that, so d+4+2=<u>d</u><u>+6</u>. August starts 31 days after that, so d+6+3=d+9, but again, we only have 7 days in our week, so this is <u>d</u><u>+2</u>. September starts 31 days after that, so d+2+3=<u>d</u><u>+5</u>. October starts 30 days after that, so d+5+2=d+7, which is just <u>d</u><u />. November starts 31 days after that, so <u>d</u><u>+3</u>. December starts 30 days after that, so <u>d</u><u>+5</u>. Remember that each one of these expressions represents a day of the week. Going back through the list (in numerical order, and listing duplicates), we have <u>d</u><u>,</u> <u>d,</u><u /> <u>d</u><u>+1</u>, <u>d</u><u>+2</u>, <u>d+3</u><u>,</u> <u>d</u><u>+3</u>, <u>d</u><u>+3</u>, <u>d</u><u>+4</u>, <u>d</u><u>+5</u>, <u>d</u><u>+5</u>, <u /><u /><u>d</u><u>+6</u><u /><u /> and <u>d</u><u>+6</u>. This means we have every day of the week covered, therefore there is a Friday the 13th at least once a year (if every day of the week can begin a month, then every day of the week can happy for any number in the month).
For leap years, every month after February would change, so we have (in the order of the months) <u></u><u>d</u>, <u>d</u><u>+3</u>, <u>d</u><u>+4</u>, <u>d</u><u />, <u>d</u><u>+2</u>, <u>d</u><u /><u>+5</u>, <u>d</u><u />, <u>d</u><u>+3</u>, <u>d</u><u /><u>+6</u>, <u>d</u><u>+1</u>, <u>d</u><u>+4</u>, a<u />nd <u>d</u><u>+</u><u /><u /><u>6</u>. We still have every day of the week represented, so there is a Friday the 13th at least once. Additionally, none of the days of the week appear more than 3 times, so there is never a year with more than 3 Friday the 13ths.<u />
I don’t think you pick an answer you just put all of those formulas and solutions into your calculator and that will give you what you need to move onto the next question/ step
Answer:
x + 9
Step-by-step explanation:
Since the divisor is in the form of <em>x - c</em>, use what is called <em>Synthetic Division</em>. Remember, in this formula, -c gives you the OPPOSITE terms of what they really are, so do not forget it. Anyway, here is how it is done:
4| 1 5 -36
↓ 4 36
----------------
1 9 0 → x + 9
You start by placing the c in the top left corner, then list all the coefficients of your dividend [x² + 5x - 36]. You bring down the original term closest to <em>c</em><em>,</em> then begin your multiplication. Now depending on what symbol your result is tells you whether the next step is to subtract or add, then you continue this process starting with multiplication all the way up until you reach the end. Now, when the last term is 0, that means you have NO REMAINDER. Finally, your quotient is one degree less than your dividend, so that 1 in your quotient can be an x, and the 9 follows right behind it, giving you the other factor of x + 9.
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