Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part III: Each correct answer will receive 4 credits. Partial credit is available


36. Write the binomial expansion of (2x − 1)⁵ as a polynomial in simplest form.

Answer:

The long way is to write out (2x - 1)(2x - 1)(2x - 1)(2x - 1)(2x - 1) and multiply. That's a long way to go for the points, but you'll get full credit.

Binomial expansion follows Pascal's Triangle. The 5th row of Pascal's Triangle is

1 5 10 10 5 1

You can find those numbers making the triangle or using ₅C_N. (Honestly, you should know 1, 5, ..., 5, 1. You should only have to check the middle numbers if you didn't know them.)

So the binomial expansion is:

(1)(2x)⁵(-1)⁰ + (5)(2x)⁴(-1)¹ + (10)(2x)³(-1)² + (10)(2x)²(-1)³ + (5)(2x)¹(-1)⁵ + (1)(2x)⁰(-1)⁶

32x⁵ - 80x⁴ + 80x³ - 40x² + 10x - 1.

The (-1)ⁿ changes the signs from plus to minus. An exponent of 0 means that that factor is 1.

37. In △ABC, m∠A = 32, a = 12, and b = 10. Find the measures of the missing angles and side of △ABC. Round each measure to the nearest tenth.

Answer:

Use the law of Sines to find the measure of ∠B. Subtract A and B from 180 to find m∠C. Use the law of Sines again to find the length of side c.

Sin A / a = Sin B / b

Sin 32 / 12 = Sin B / 10

Sin B = 10 (Sin 32) / 12

Sin B = 0.4416

B = 26.2 degrees

m∠C = 180 - (32 + 26.2) = 121.8

Sin A / a = Sin C / c

Sin 32 / 12 = Sin 121.8 / c

c = 12 (Sin 121.8) / Sin 32

c = 19.24 = 19.2

38. The probability that the Stormville Sluggers will win a baseball game is 2/3. Determine the probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games.

Answer:

The probability that they will win at least 6 games is the probability that they will win six games plus the probability that they will win seven games plus the probability that they will win eight games.

P(6 wins) = ₈C₆(2/3)⁶(1/3)² = 0.273129...

P(7 wins) = ₈C₇(2/3)⁷(1/3)¹ = 0.1560737...

P(8 wins) = ₈C₈(2/3)⁸(1/3)⁰ = 0.039018...

0.27312 + 0.15607 + 0.03901 = 0.4682 = 0.468.

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.