Упростите выражения! (m^3 - 5m +2) : (m^3+2m^2+m)

July 2022 1 20 Report

Упростите выражения!
(m^3 - 5m +2) : (m^3+2m^2+m) - дробь это
(a^4+a^3+a^2+a+1) : (a^5-1) - дробь
(x^2(y+1)-y^2(x+1) ) : (x(y+1)^2-y(x+1)^2) - дробь

Answers & Comments

90misha90

Verified answer

$x^n-y^n=$

$(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+...+x^2y^{n-3}+xy^{n-2}+y^{n-1})$ , где $n\in N$

$\frac{a^4+a^3+a^2+a+1}{a^5-1}= \frac{a^4+a^3+a^2+a+1}{a^5-1^5}= \frac{a^4+a^3+a^2+a+1}{(a-1)(a^4+a^3*1+a^2*1^2+a*1^3+1^4)}=$

$=\frac{a^4+a^3+a^2+a+1}{(a-1)(a^4+a^3+a^2+a+1)}= \frac{1}{a-1}$

---------------------------------

$\frac{m^3-5m+2}{m^3+2m^2+m}= \frac{m^3-5m+2}{m(m^2+2m+1)}= \frac{m^3+2m^2-m-2m^2-4m+2}{m(m+1)^2}=$

$= \frac{m(m^2+2m-1)-2(m^2+2m-1)}{m(m+1)^2} = \frac{(m-2)(m^2+2m-1)}{m(m+1)^2}$

как видно, дробь оказалась не сокращаемой

скорее всего в условии ошибка в знаменателе и было бы тогда так:

$\frac{m^3-5m+2}{m^3+2m^2-m}= \frac{m^3-5m+2}{m(m^2+2m-1)}= \frac{m^3+2m^2-m-2m^2-4m+2}{m(m^2+2m-1)}=$

$= \frac{m(m^2+2m-1)-2(m^2+2m-1)}{m(m^2+2m-1)} = \frac{(m-2)(m^2+2m-1)}{m(m^2+2m-1)}=\frac{m-2}{m}=1-\frac{2}{m}$

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$\frac{x^2(y+1)-y^2(x+1)}{x(y+1)^2-y(x+1)^2} = \frac{x^2y+x^2-xy^2-y^2}{x(y^2+2y+1)-y(x^2+2x+1)}=$

$= \frac{x^2y-xy^2+x^2-y^2}{xy^2+2xy+x-x^2y-2xy-y} = \frac{xy(x-y)+(x+y)(x-y)}{xy^2+x-x^2y-y}=$

$= \frac{[xy+(x+y)]*(x-y)}{xy^2-x^2y+x-y} = \frac{(xy+x+y)(x-y)}{(-xy)*(x-y)+1*(x-y)} = \frac{(xy+x+y)(x-y)}{[(-xy)+1]*(x-y)}=$

$= \frac{xy+x+y}{-xy+1}= \frac{xy+x+y}{1-xy}$

Answers & Comments

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