Search for: "factoring"

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Keywords: TopicsProblem Features Limits - unbounded - limits at infinity - trigonometric limits . . . - differential equations - domain & range - factoring - finding roots - fixed point - geometry . . .
3K - last updated 2009-08-20 23:03 by EricHsu

factoring
Factoring most famously comes up in (a) limits of fractions with canceling terms and (b) partial fractions. . . . early in the year, students in my class found factoring polynomials something to bond over. In particular, . . . find revolutionizes their relationship with factoring. (Unmultiplying is essentially short division . . . sometime in the semester. Some people think that factoring is one of the skills that will disappear . . .
3K - last updated 2009-08-20 23:03 by EricHsu

1. determine if the answer is correct;
2. determine if there are any mistakes made in solving the problem and list them ({note} that just because the answer is correct does not mean there are no mistakes);
3. if there are mistakes, redo the problem correctly; if there are no mistakes, devise {ANOTHER} correct method to solve the problem.
   1. \( \frac{x^2-1}{ x+1}=\frac{x^2+(-1)}{ x+1}=\frac{x^2}{ x}+\frac{-1}{1}=x-1                               \>  \)
   2. \( (x+y)^2-(x-y)^2=x^2+y^2-x^2-y^2=0 \)
   3. \( \frac{9(x-4)^2}{3x-12}=\frac{3^2(x-4)2}{3x-12}=\frac{(3x-12)^2}{3x-12} =3x-12 \)             
   4. \( \frac{x^2y^5}{2x^{-3}}=x^2y^5\cdot2x^3=2x^6y^5 \)
   5. \( \frac{(2x^3+7x^2+6)-(2x^3-3x^2-17x+3)}{(x+8)+(x-8)}=\frac{(2x)^2-17x+9}{2x} =2x-17x+9=-15x+9=-6x \)
   6. \( \frac{x^{-1}+y^{-1}}{ x^{-1}-y^{-1}}=\frac{(x+y)^{-1}}{(x-y)^{-1}} =\left(\frac{x+y}{ x-y}\right)^{-1}=-\frac{x+y}{ x-y}=\frac{x+y}{ y-x} \)

1. \( \lim_{x\to1}\frac{x^3-5x^2+8x-4}{ x^3-x^2-4x+4} \)
2. \( \lim_{x\to3^-}\frac{|x-3|}{9-x^2} \)
3. \( \lim_{x\to2}\frac{1}{ x-2}-\frac{4}{ x^2-4} \)
4. \( \lim_{x\to-1}\frac{x^4+x}{ x^2+x} \)