Question

Divide and simplify these examples of algebric
fractions
a) 2x-2/4 divide x²-2x+ 1/6
b) (p-1)^2/1-p divide (1-p)+p(p-1)/1-p^3
c) px^2-py^2/x^2-2xy+y^2 divide p^2y-p^2x/(x-y) ^2
d) 1-2p+p^2/mp^2-m divide m-mp^3/(m^2p+m^2)(1+p^2+p)


please answer this question
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Answer 1

${\huge{\underline{\boxed{\red{\mathcal{Answer}}}}}}$

$\bf{a) \dfrac{2x-2}{4} ÷ \dfrac{x^2-2x+1}{6}} \\\\\implies \dfrac{x-1}{2} ÷ \dfrac{(x-1)^2}{6} \\\\\implies \dfrac{\dfrac{x-1}{2}}{\dfrac{(x-1)^2}{6}} \\\\\bf{\implies \dfrac{3}{x-1}}$

$\bf{b) \dfrac{(p-1)^2}{1-p} ÷ \dfrac{(1-p)+p(p-1)}{1-p^3}} \\\\\implies \dfrac{(p-1)^2}{-(p-1} ÷ \dfrac{1-p+p^2-p}{1-p^3} \\\\\implies -(p-1) ÷ \dfrac{p^2+1-2p}{1-p^3} \\\\\implies 1-p ÷ \dfrac{(1-p)^2}{(1-p)(1+p+p^2)} \\\\\implies 1-p ÷ \dfrac{1-p}{1+p+p^2} \\\\\implies \dfrac{\dfrac{1-p}{1}}{\dfrac{1-p}{1+p+p^2}} \\\\\bf{\implies 1+p+p^2}$

$\bf{c) \dfrac{px^2-py^2}{x^2-2xy+y^2} ÷ \dfrac{p^2y-p^2x}{(x-y)^2}} \\\\\implies \dfrac{p(x^2-y^2)}{(x-y)^2} ÷ \dfrac{p^2(y-x)}{(x-y)^2} \\\\\implies \dfrac{p(x-y)(x+y)}{(x-y)^2} ÷ \dfrac{-p^2(x-y)}{(x-y)^2} \\\\\implies \dfrac{p(x+y)}{x-y} ÷ \dfrac{-p^2}{x-y} \\\\\implies \dfrac{\dfrac{p(x+y)}{x-y}}{\dfrac{-p^2}{x-y}} \\\\\bf{\implies \dfrac{-(x+y)}{p} }$

$\bf{d) \dfrac{1-2p+p^2}{mp^2-m} ÷ \dfrac{m-mp^3}{(m^2p+m^2)(1+p^2+p)}} \\\\\implies \dfrac{(p-1)^2}{m(p^2-1)} ÷ \dfrac{m(1-p^3)}{m^2(p+1)(1+p+p^2)} \\\\\implies \dfrac{(p-1)^2}{m(p-1)(p+1)} ÷ \dfrac{(1-p)(1+p^2+p)}{m(p+1)(1+p+p^2)} \\\\\implies \dfrac{\dfrac{p-1}{m(p+1)}}{\dfrac{-(p-1)}{m(p+1)}} \\\\\bf{\implies -1}$

Algebraic Identities:

$\boxed{\begin{minipage}{7 cm}\boxed{\bigstar\:\:\textbf{\textsf{Algebric\:Identity}}\:\bigstar}\\\\1)\bf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\2)\bf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\3)\bf\: A^{2} - B^{2} = (A+B)(A-B)\\\\4)\bf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\5)\bf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\6)\bf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\7)\bf\:(A-B)^{3} = A^{3} - 3AB(A-B) - B^{3}\\\\8)\bf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2})\\\\9)\bf\: A^{3} - B^{3} = (A-B)(A^{2} + AB + B^{2})\\\\ \end{minipage}}$

Hope it helps!

Answer 2

refwr to the attachment

thx for points.