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Vaselesa [24]
3 years ago
7

The product of three consecutive integers is -1716. what is the greatest of the three integers?

Mathematics
1 answer:
Monica [59]3 years ago
6 0

Answer:

-11

Step-by-step explanation:

If we call x the least integer, then x + 1 is the next integer, and x + 2 is the greatest integer.

x (x + 1) (x + 2) = -1716

x (x² + 3x + 2) = -1716

x³ + 3x² + 2x = -1716

x³ + 3x² + 2x + 1716 = 0

That's not an easy equation to factor.  We could use rational root test, but that would take a long time.  However, we can narrow down the options logically.  The three integers are relatively close together, so if we approximated the product as x³:

x³ ≈ -1716

x ≈ -11.9

So let's see if x = -12 is a solution:

(-12)³ + 3(-12)² + 2(-12) + 1716 = 396

Nope.  Let's try x = -11:

(-11)³ + 3(-11)² + 2(-11) + 1716 = 726

Not that either.  Let's go the other direction, x = -13:

(-13)³ + 3(-13)² + 2(-13) + 1716 = 0

Aha!  x = -13.  So x + 1 = -12, and x + 2 = -11.

-11 × -12 × -13 = -1716

The greatest integer is -11.

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2 years ago
Write the complex number -2+2i in polar form
FromTheMoon [43]
For polar form you need to find the modulus (length of the vector) and the argument (angle of the vector) and present in form rcis(Arg) or re^Argi

start with the modulus r=sqrt(a^2 +b^2)
                                     =sqrt(-2^2 +2^2)
                                     = sqrt(4+4)
                                     =sqrt(8)
                                     =2sqrt(2)

next the argument, firstly arg=tan(b/a)
                                              = tan(2/2)
                                              =tan(1)
                                             =pi/4 .     (exact values table)
Now consider the quadrant the complex number is in, as it is (-2,2) it is in the second quadrant and as such your Arg value is:
Arg=pi-arg
     = pi-pi/4
     = 3pi/4

add it all together and your complex number in polar form is:
2sqrt2cis(3pi/4)

note: cis is short hand for cos(x)+isin(x), it is possible your tutor would rather you use the complex exponential form which is simply re^Argi and your answer would look like:
2sqrt2e^(3pi/4)i

Also notice the difference between arg and Arg as this often slips students up and always present Arg in prinicple argument form ie -pi<Arg<pi

Hopefully this has been clear enough and good luck

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