Problem Set Suggestions

doughishere

doughishere

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I want to work on my more advanced factoring (its probably not more advanced). Anyone have any suggestions on problem sets that are similar to e) and f)?


Just like 30-40 that are similar or any really.



TIA!
 

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There are many odd things about how math is taught. I am not sure that there are many real-world problems that require factoring. Factoring is usually a convenience rather than a necessity. And if finding a factoring takes a lot of time, then it is not even a convenience. Frequently (but not always) a common factor in each term provides the clue if there is a simple factorization. (There is no guarantee that there is any factorization of an expression unless it is a polynomial, and there is no guarantee that the factoring will be simple even if the expression is factorable.)

Let's start with x3y−4xy.\displaystyle x^3y - 4xy.

It should be obvious that both x and y are factors of every term. So

x3y−4xy=xy(x2−4).\displaystyle x^3y - 4xy = xy(x^2 - 4). Now you should see a difference of like powers.

x3y−4xy=xy(x2−4)=xy(x−2)(x+2).\displaystyle x^3y - 4xy = xy(x^2 - 4) = xy(x - 2)(x + 2).

3x3/2−9x1/2+6x−1/2.\displaystyle 3x^{3/2} - 9x^{1/2} + 6x^{-1/2}.

This one is a lot harder. It is obvious that 3 is a common factor in each term, but it may be harder to see that x−1/2\displaystyle x^{-1/2} is also a common factor.

x3/2=x4/2∗x−1/2=x2∗x−1/2 and x1/2=x2/2∗x−1/2=x∗x−1/2.\displaystyle x^{3/2} = x^{4/2} * x^{-1/2} = x^2 * x^{-1/2} \text { and } x^{1/2} = x^{2/2} * x^{-1/2} = x * x^{-1/2}.

∴3x3/2−9x1/2+6x−1/2=3(x3/2−3x1/2+2x−1/2)=\displaystyle \therefore 3x^{3/2} - 9x^{1/2} + 6x^{-1/2} = 3(x^{3/2} - 3x^{1/2} + 2x^{-1/2})=

3x−1/2(x2−3x+2)=3(x−2)(x−1)x=3(x−2)(x−1)xx.\displaystyle 3x^{-1/2}(x^2 - 3x + 2) = \dfrac{3(x - 2)(x - 1)}{\sqrt{x}} = \dfrac{3(x - 2)(x - 1)\sqrt{x}}{x}.
 
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