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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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More than anything she wanted the world to be uncomplicated, for right and wrong to be as easily divided as the black and white sections of an Oreo. But the world was not a cookie.
— Roshani Chokshi, Aru Shah and the Tree of Wishes