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Emile Kessler

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Q: For any number c the polynomial x2 - c2 can be factored as?
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How do you figure out whether 999919 is a prime number or not using only paper-pen calculations in few minutes?

To test primality, you need to check whether the number is divisible by the prime numbers less than the integer part of its square root. I did it by noticing that 999,919 is almost 1,000,000 when I was determining what number would be close to its square root. The square root of 1,000,000 is 1,000. Then, since 999,919 is 1,000,000 - 81, it is one square number minus another: 1,0002 - 92. So, I know that 999,919 can be factored because of the Pythagorean theorem: a2 + b2 = c2, which can be rewritten as a2 = c2 - b2 = (c - b)(c + b). By substituting 999,919 for a2, 9 for b, and 1000 for c, I have 999,919 = (1000 - 9)(1000 + 9) = 991 x 1009. So, because 999,919 has factors other than 1 and itself, it cannot be a prime number.


Simplify 8 to the 8 power times 8 neg power?

a2+b2=c2 a=24+b=24=c2 576=576 ------------ 2 576


Factor c4 c3 - c2 - c?

If that's + c3, the answer is c(c - 1)(c + 1)(c + 1)


Are irrational numbers cannot be represented on a number line?

Irrational numbers can be represented on a number line. For example, to graph the square root of two, draw a line of 1 unit (1 unit = the distance between the points of two whole numbers) from -1 which is perpendicular to the number line. Then, using a compass, place the pointy end on 0, the pencil tip on the end of the drawn line that is not touching the number line and drawing an arc so that it hits the number line on the positive side. Draw a point at where the arc meets the number line. That point is the square root of 2. This works because of Pythagoras theorem (a2+b2=c2, 12+12=22).


How do you figure out the slope of grade in percent?

1. Determine the height difference (a) between the start and end of the slope by subtracting the starting height from the final height (if it is a downhill slope the difference will be a negative number) 2. Determine the horizontal distance (b) between the starting point and the final point. If you only know the actual linear distance (c) along the slope, you can figure the horizontal distance using the Pythagorean Theorem: a2+b2=c2. For example, if the linear distance (c) along the slope from start to finish is 1000 feet, and the height difference (a) is 75 feet, then using the theorem you would have 752+b2=10002or 5625+b2=1000000. Your horizontal distance (b) would therefore be b2=994375 or b=997.2 feet 3. Divide the height difference (a) by the horizontal distance (b) and multiply by 100. This gives you the grade of the slope in percent. In the example it would be 75 ÷ 997.2 × 100 = 7.5% grade