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Identities, Polynomials...this is why I hate math.



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Thu Aug 19, 2010 3:21 pm
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iceprincess says...



All of these questions below are supposed to be brain teasers/summer holiday homework! Can someone please help? :(

1) Suppose x(x + a) - 10 = (x - b)(x + c). Suggest a possible set of values of a, b and c.

2) For the expression ax^2 + bx + 25, write down two possible sets of the values of a and b such that the expression is a perfect square.

3) Suggest two ways of factorising x^6 - y^6.

4) Find two fractions with different denominators such that their difference is 2x -y/4xy.

There are actually 20 questions like this, but I've already figured most of them out, except for these. I would really appreciate it if you guys helped me out :D
you'll never find another sweet little girl with sequined sea foam eyes
ocean lapping voice, smile coy as the brightest quiet span of sky
and you're all alone again tonight; not again, not again, not again.
and don't it feel alright, and don't it feel so nice? lovely.


  





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Tue Aug 24, 2010 10:50 pm
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Kale says...



For #1, I distributed the x on the left to get x^2 + ax - 10. Factoring that gets you the sets (x + 5)(x - 2) and (x + 10)(x - 1), which result in a being either 3 or 9 respectively. I then expanded the right side to get x^2 + bx + cx - bc.

bc = -10
bx + cx = ax

For #2, a must be a number that has a square root. Depending on what you choose for a, b will be affected while 25 remains constant. I'll be using c to represent the square root of a. And so the formula for the factored expression would be (cx + 5)^2. Pick a value for c, and expand to find a and b.

For #3, I'm not sure what they want, but I think it might be factoring starting from (x^3)^2 - (y^3)^2 or (x^2)^3 - (y^2)^3.

For #4, it's a matter of finding two denominators that have 4xy as their least common multiple whose numerators have a difference of 2x - y. It's easiest to work this one backwards, setting all the denominators to 4xy, and making sure each numerator has a term that can be canceled (I hope that made sense).

Basically:

(2x - y)/4xy = (a)/4xy - (b)/4xy

where a and b are both divisible by the terms 4, x, or y, though both a and b cannot be divisible by the same term. So if a is divisible by 4, b must be divisible by either x or y.

You can cheat and have one of the denominators be 4xy. :P
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