Quandaries and Queries


Who is asking: Student
Level: Secondary

I'm in FST and I am so confused and we have a test coming up soon. I have these practice problems that I don't know what to do, and I was hoping someone could help me out. I put what I think it should be next to it, so you mabye can tell me whats wrong, if anything.
Here they are:

Factor these:
x3+125 -----> (x+5)3
8x3-27 -----> (?)
x2+36 -----> (x+6)2
x4-5x2+4 --> (?)

Any shortcuts are appreciated. I'm usually pretty good in math, but I just can't grasp this, I just don't get it.




Hi KJ,

There are some patterns that you need to recognize to in order to factor. Three if these patterns are:

the difference of squares

a2 - b2 = (a - b)(a + b)

the difference of cubes

a3 - b3 = (a - b)(a2 + ab + b2)

and the sum of cubes

a3 + b3 = (a + b)(a2 - ab + b2).

For example, in your first problem, 125 = 53 and thus this is a sum of cubes

x3+125 = x3 + 53 = (x + 5)( x2 - 5x + 25)

In a similar way your second problem is a difference of cubes.

Problem 3 doesn't match any of the three paterns above, in fact the exprerssion in problem 3 can't be factored.

Problem 4 is again recognizing a pattern. When I look at problem 4 I see

(quantity 1) - 5 (quantity 2) + 4


(quantity 1) is the square of (quantity 2)

To be more formal. Let y = x2 then

x4-5x2+4 = y2 -5y + 4

Factor this expression then replace y by x2 and see if the resulting expression can be further factored.



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