Algebra Problems of the Day (Integrated Algebra Regents, June 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, June 2011

Part I: Each correct answer will receive 2 credits.


1. The expression x² - 36y² is equivalent to

1) (x - 6y)(x - 6y)
2) (x - 18y)(x - 18y)
3) (x + 6y)(x - 6y)
4) (x + 18y)(x - 18y)

Answer: 3) (x + 6y)(x - 6y)

When we have the Difference of Two Perfect Squares, it can be factored into two binomials which are conjugates. Take the sqaure roots of each of the term.

That is choice (3).

Choice (1) would be x² - 12xy + 36y².

Choices (2) and (4) have 18, which is half of 36, instead of 6, which is the square root.

2.The legs of an isosceles right triangle each measure 10 inches. What is the length of the hypotenuse of this triangle, to the nearest tenth of an inch?

1) 6.3
2) 7.1
3) 14.1
4) 17.1

Answer: 3) 14.1

You can use Pythagorean Theorem to find the answer if you didn't know that the hypotenuse in an right isosceles triangle is the legnth of one leg times SQRT(2).

10 * 1.414... = 14.14...

3. The expression (12w⁹y³) / (-3w³y³) is equivalent to

1) -4w⁶
2) -4w³y
3) 9w⁶
4) 9w³y

Answer: 1) -4w⁶

Divide the coeffients and subtract the exponents of the like variables.

12/(-3) = -4, so eliminate Choices (3) and (4).

w⁹ / w³ = w9 - 3 = w6. This is Choice (1).

4. The spinner shown in the diagram below is divided into six equal sections.

1) an odd number
2) a prime number
3) a perfect square
4) a number divisible by 2

Answer: 3) a perfect square

Which category has the fewest members n the group of integers from 1 to 6?

There are 3 odd numbers.

There are 3 prime numbers.

There are 2 perfect sqaures.

There are 3 numbers divisible by 2 (even numbers).

So perfect squares are the least likely.

5. What are the factors of the expression x² + x - 20?

1) (x + 5) and (x + 4)
2) (x + 5) and (x - 4)
3) (x - 5) and (x + 4)
4) (x - 5) and (x - 4)

Answer: 2) (x + 5) and (x - 4)

The -20 tells us that the factors have different signs. The + x tells us that the larger factor is has a plus sign. That is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

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