Eliminate. Terminate. Totally, completely Annihilate.
Someone obviously used the Difference of Square rule in this example.
I wanted to do this before the Algebra Regents exams when it might have been more helpful to some students, but the fact is, they don't actually follow me. They just find the posts with the questions and answers on them through search engines.
The Difference of Squares is a common topic for Algebra 1, but what about other powers, such as cubes and quarts? (Quarts are is "quartic", fourth power.)
In a binomial that looks like x2 - n2, where n is any number, which will be squared, we can factor the two terms into conjugates:
All nice and neat. When you multiply the two conjugates (which have the same terms, but one has a plus and one has a minus), the two x terms -- (nx) and (-nx) -- are additive inverses and will sum to zero.
Every now and then, state exams like to throw a curve ball at you, and use a higher power, usually 4.
What if the problem said:
So, if they really wanted to be mean, er, I mean "challenging", they could go another step further and ask:
I know that I started by saying "difference of ...", but that was to keep everything flowing.
Unlike squares (and other even powers), you can factor the sum of two cubes. Also unlike squares, a perfect cube can be a negative number, so it also could be written as a difference of cubes. So I'm not wrong -- I just renamed it for no reason.
There's a simple format for the Sum of Two Cubes:
First thing you should see is that (x + n) means that there is a root at x = -n.
The second thing you should see is that that is the only real root. If you check the discriminant, b2 - 4ac, you get the following:
(n)2 - 4(1)(n2).
As long as n has a nonzero value, the discriminant will always be less than zero. This example had a leading coefficient of 1, but it is still true if there is a different number in front the x3 term.
If we wanted to put this together, we could try to factor x6 - 64.
Or really put it all together and got with an x12 term. But who wants to do that?
25. In the equation x2 + 10x + 24 = (x + a)(x + b), b is an integer. Find algebraically all possible values of b.
x2 + 10x + 24 factors into (x + 4)(x + 6), the order of the two binomials doesn't matter.
So the possible values of b are 4 and 6.
26. Rhonda deposited $3000 in an account in the Merrick National Bank, earning 4.2% interest, compounded annually. She made no deposits or withdrawals. Write an equation that can be used to find B, her account balance after t years.
The formula you need is B = P(1 + r)t
P = 3000, the principal, or starting amount
r = .042, which is 4.2% as a decimal
t is time. No specific value is given, so use the variable.
That makes the answer B = 3000(1.042)t
Any questions?
If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.
34. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters, as shown in the diagram below. Together the walkway and the garden have an area of 396 square meters.
Write an equation that can be used to find x, the width of the walkway.
Describe how your equation models the situation.
Determine and state the width of the walkway, in meters.
Unlike the previous question, this was a majorly complicated problem, with more binomial multiplication.
The length of the rectangle is x + 16 + x or (2x + 16).
The width of the rectangle is x + 12 + x or (2x + 12).
The area of the rectangle is 396.
We can discard the negative value not because we made a mistake, but because a negative value makes no sense in the context of length and width.
You are left with the width of the walkway being 3 meters wide.
The equation models the situation because the Area is the product of the length times the width. The length of the garden plus the walkway on either side is 2x + 16. The width of the garden plus the walkway is 2x + 12. The Area of the garden is 396 square meters.
Any questions?
If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.
30. The function f has a domain of {1, 3, 5, 7} and a range of {2, 4, 6}. Could f be represented by {(1, 2), (3, 4), (5, 6), (7, 2)}? Justify your answer.
I'm still not happy about the wording of this one. I've been told that I'm "splitting hairs", but, in my opinion, the question is asked backward.
The question is asking if the set given could be function f. In other words, is it a function and does it have the correct domain and range?
Let's answer that question.
The set of ordered pairs is a function because no x value is mapped to more than one y value. The domain (the x values) are {1, 3, 5, 7} as in f. The range (the y values) are {2, 4, 6} which matches the range of f. So, yes, it could represent f.
Remember that "yes" or "no" without an explanation is worth ZERO points.
31. Factor the expression x4 + 6x2 - 7 completely.
Usually, a “factor completely” question has a Greatest Common Factor (GCF) component to it. This one doesn't, but it does have an extra step that isn't obvious at first.
If you are thrown off by the exponents being 4 and 2, instead of 2 and 1, that's okay. They will act pretty much the same way.
x4 + 6x2 - 7 factors into (x2 + 7) (x2 - 1). You now have two binomials, each with an x2 term. If you look closely, you can see that we aren't finished. We can factor further.
Using the Difference of Squares Rule (x2 - 1) can be factored into (x – 1)(x + 1).
This makes the final answer:
Any questions?
If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.
