Algebra Problems of the Day (Integrated Algebra Regents, January 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, January 2011

Part I: Each correct answer will receive 2 credits.


26. What is the result when 2x² + 3xy − 6 is subtracted from x₂ − 7xy + 2?

1) -x² - 10xy + 8
2) x² + 10xy − 8
3) -x² - 4xy − 4
4) x² - 3xy − 4

Answer: 1) -x² - 10xy + 8

The "from" goes on top and the other expression goes on the bottom.

x₂ − 7xy + 2
2x² + 3xy − 6

-x² - 10xy + 8

This is Choice (1).

Watch your signs when subtracting negatives.

27. What is an equation of the axis of symmetry of the parabola represented by y = −x² + 6x − 4?

1) x = 3
2) y = 3
3) x = 6
4) y = 6

Answer: 1) x = 3

The Axis of Symmetry is ALWAYS x = a number. Eliminate Choices (2) and (4).

The formula is x = -b/(2a), where a = -1 and b = 6.

So x = -(6)/(2(-1)) = -6/-2 = 3, which is Choice (1).

They didn't even try to trick you with x = -3 in case you forgot a sign.

28. Which equation has roots of −3 and 5?

1) x² + 2x - 15 = 0
2) x² - 2x - 15 = 0
3) x² + 2x + 15 = 0
4) x² -+ 2x + 15 = 0

Answer: 2) x² - 2x - 15 = 0

If the roots are -3 and 5, then multiply (x + 3)(x - 5).

(x + 3)(x - 5) = x² - 5x + 3x - 15 = x² - 2x - 15, which is Choice (2).

29. A spinner that is equally divided into eight numbered sectors is spun 20 times. The table below shows the number of times the arrow landed in each numbered sector.

Based on the table, what is the empirical probability that the spinner will land on a prime number on the next spin?

1) 9/20
2) 11/20
3) 12/20
4) 14/20

Answer: 3) 12/20

The prime numbers on the spinner are 2, 3, 5, and 7. (1, 4, 6, and 8 are NOT prime numbers.) Those four numbers appeared 3, 2, 4, and 3 times, respectively.

The sum of 3 + 2 + 4 + 3 = 12, so the empirical probability is 12/20, which is Choice (3).

30. Which expression represents (x² - x - 6) / (x² − 5x + 6) in simplest form?

1) (x + 2) / (x - 2)
2) (-x - 6) / (-5x + 6)
3) 1/5
4) 1

Answer: 1) (x + 2) / (x - 2)

Since the numerator does NOT match the denominator, it should be obvious that the answer is NOT 1. Eliminate Choice (4).

Factor both the numerator and the denominator into two binomials. Cancel out the common pairs which make 1.

(x² - x - 6) factors into (x - 3)(x + 2).

(x² − 5x + 6) factors into (x - 3)(x - 2)

(x - 3)/(x - 3) = 1

So the fraction reduces to (x + 2)/(x - 2), which is Choice (1).

More to come. Comments and questions welcome.

More Regents problems.

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