## Tuesday, May 08, 2018

### Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

August 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.

19. Consider the system shown below.

2x − y = 4
(x + 3)2 + y2 = 8

The two solutions of the system can be described as

1) both imaginary
2) both irrational
3) both rational
4) one rational and one irrational

Rewrite the first equation as y = 2x - 4, then substitute it into the second equation:

(x + 3)2 + (2x - 4)2 = 8
x2 + 6x + 9 + 4x2 - 16x + 16 = 8
5x2 - 10x + 25 = 8
5x2 - 10x + 17 = 0

a = 5, b = -10, c = 17
Discriminant: b2 - 4ac = (-10)2 - 4(5)(17) = -240, which is less than 0.
Roots are imaginary.

Had the discriminant been positive, you could have continued using the Quadratic Formula to determine if the roots were both rational or both irrational. One of each wouldn't happen.

20. Which binomial is not a factor of the expression x3 − 11x2 + 16x + 84?

1) x + 2
2) x + 4
3) x - 6
4) x - 7

A quick graph of x3 − 11x2 + 16x + 84 shows zeroes at -2, 6, and 7, meaning that (x + 2), (x - 6), and (x - 7) are factors.

If you want to work it out using division, you will see that (x + 4) leaves a remainder, so it is not a factor of the polynomial. The other three do not leave remainders. (see image below)

21. A ball is dropped from a height of 32 feet. It bounces and rebounds 80% of the height from which it was falling. What is the total downward distance, in feet, the ball traveled up to the 12th bounce?

1) 29
2) 58
3) 120
4) 149

If it falls from 32 feet, how could the distance be 29 feet? Twelve bounces later, 58 is "obviously" incorrect as well. Eliminate those two before you do anything else.

The total distance up until that the nth bounce is
Sn = (32 - 32(.8)12) / (1 - .8) = 149.004883722... = 149