Algebraické výrazy

1. Algebraické výrazy a jejich úpravy

1)

$\huge\frac{{1-}\frac{x}{x+2}}{\frac{x}{x+2}{+1}}$

$\huge\frac{{1-}\frac{x}{x+2}}{\frac{x}{x+2}{+1}} = \frac{\frac{x+2-x}{x+2}}{\frac{x+x+2}{x+2}} = \frac{\frac{2}{x+2}}{\frac{2x+2}{x+2}} = \frac{\frac{2}{x+2}}{\frac{2(x+1)}{x+2}} = \frac{2}{x+2}\times \ \frac{x+2}{2(x+1)} = \frac{1}{x+1} \\x\neq-1, x\neq-2$

2)

$\huge\bigg(\frac{x^3}{y^2}+\frac{x^2}{y}+x+y\bigg)\div \bigg(\frac{x^2}{y^2}-\frac{y^2}{x^2}\bigg) $


$\huge\bigg(\frac{x^3}{y^2}+\frac{x^2}{y}+x+y\bigg)\div \bigg(\frac{x^2}{y^2}-\frac{y^2}{x^2}\bigg) = \frac{x^3+x^2y+xy^2+y^3}{y^2}\div \frac{x^4-y^4}{x^2y^2} =\\ \huge\frac{x^2(x+y)+y^2(x+y)}{y^2} \times \frac{x^2y^2}{(x^2 - y^2)(x^2+y^2)} = \frac{(x^2+y^2)(x+y)}{y^2} \times \frac{x^2y^2}{(x^2 - y^2)(x^2+y^2)}=\\ \huge\frac{x^2(x+y)}{(x-y)(x+y)} = \frac{x^2}{x-y} \\ \\x\neq0, y\neq0, x\neq+1,y\neq-1 $

3)

$\huge\left[\left(\frac{n+2}{n-2}\right)^3 \div \frac{n^3+4n^2+4n}{3n^2-12n+12}\right]\times \frac{n}{3} $


$\huge\left[\left(\frac{n+2}{n-2}\right)^3 \div \frac{n^3+4n^2+4n}{3n^2-12n+12}\right]\times \frac{n}{3}= $ $\huge\left[\frac{(n+2)^3}{(n-2^3} \div \frac{n(n^2+4n+4)}{3(n^2-4n+4)}\right]\times \frac{n}{3} =$ $\huge\left[\frac{(n+2)^3}{(n-2^3} \div \frac{n(n+2)^2}{3(n-2)^2}\right]\times \frac{n}{3} =$ $\huge\frac{(n+2)^3}{(n-2)^3} \times \frac{3(n-2)^2}{n(n+2)^2}\times \frac{n}{3} =\frac{n+2}{n-2}$ $n\neq0, n\neq+2, n\neq-2$

4)

$\huge\frac{a^4-b^4}{a^2b^2}\div \left [(1+\frac{b^2}{a^2})\times (1-\frac{2a}{b}+\frac{a^2}{b^2}) \right]$


$\huge\frac{a^4-b^4}{a^2b^2}\div \left [(1+\frac{b^2}{a^2})\times (1-\frac{2a}{b}+\frac{a^2}{b^2}) \right]=$ $\huge\frac{(a^2-b^2)(a^2+b^2)}{a^2b^2}\div \left [\frac{a^2+b^2}{a^2}\times \frac{b^2-2ab+a^2}{b^2} \right]=$ $\huge\frac{(a^2-b^2)(a^2+b^2)}{a^2b^2}\div \left [\frac{(a^2+b^2)\times (a-b)^2}{a^2b^2} \right]=$ $\huge\frac{(a^2-b^2)(a^2+b^2)}{a^2b^2}\times \frac{a^2b^2}{(a^2+b^2)\times (a-b)^2} =$ $\huge\frac{(a^2-b^2)}{(a-b)^2} = \frac{(a-b)(a+b)}{(a-b)(a-b)} = \frac{a+b}{a-b} \\a\neq b, a\neq0, b\neq0$

5)

$\huge 2a-\left(\frac{2a-3}{a+1}-\frac{a^2+3}{2a^2-2}-\frac{a+1}{2-2a}\right) \times \frac{a^3+1}{a^2-2}$

$\huge2a-\left(\frac{2a-3}{a+1}-\frac{a^2+3}{2(a^2-1)}-\frac{a+1}{2(1-a)}\right) \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\left(\frac{2a-3}{a+1}-\frac{a^2+3}{2(a-1)(a+1)}-\frac{a+1}{2(1-a)}\right) \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\left(\frac{2(a-1)(2a-3)-a^2-3+(a+1)(a+1)}{2(a+1)(a-1)}\right) \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\left(\frac{2(a^2-3a-2a+3)-a^2-3+a^2+2a+1}{2(a+1)(a-1)}\right) \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\left(\frac{4a^2-10a+6-2+2a}{2(a+1)(a-1)}\right) \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\frac{4a^2-8a+4}{2(a+1)(a-1)}\times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\frac{4(a^2-2a+1)}{2(a+1)(a-1)} \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\frac{4(a-1)^2}{2(a+1)(a-1)} \times \frac{a^3+1}{a(a-1)}=$

$\huge2a-\frac{2(a^3+1)}{a(a+1)}=\frac{2a^2(a+1)-2(a^3+1)}{a(a+1)}=$

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

$\huge\frac{2(a+1)(a-1)}{a(a+1)}=\frac{2(a-1)}{a}$


$x\neq-1$

$x\neq0$

6)

$\huge\left(\frac{1}{a+1}-\frac{2a}{a^2-1}\right)\times\left(\frac{1}{a}-1\right)$


$\huge\left(\frac{1}{a+1}-\frac{2a}{a^2-1}\right)\times\left(\frac{1}{a}-1\right)=$ $\huge\frac{(a-1)-2a}{(a+1)(a-1)}\times\frac{1-a}{a}=$ $\huge\frac{a-1-2a}{(a+1)(a-1)}\times\frac{-(a-1)}{a}=$ $\huge\frac{-a-1}{(a+1)(a-1)}\times\frac{-(a-1)}{a}=\frac{-(a+1)}{(a+1)(a-1)}\times\frac{-(a-1)}{a}=\frac{1}{a} \\a\neq0, a\neq-1, a\neq+1$

7)

$\huge\frac{a^2+ab}{a^2+b^2}\times \left(\frac{a}{a-b}-\frac{b}{a+b}\right)$


$\huge\frac{a^2+ab}{a^2+b^2}\times \left(\frac{a}{a-b}-\frac{b}{a+b}\right)=$ $\huge\frac{a(a+b)}{a^2+b^2}\times \left(\frac{a(a+b)-b(a-b)}{(a-b)(a+b)}\right)=$ $\huge\frac{a(a+b)}{a^2+b^2}\times \left(\frac{a^2+ab-ab+b^2}{(a-b)(a+b)}\right)=$ $\huge\frac{a(a+b)}{a^2+b^2}\times \left(\frac{a^2+b^2}{(a-b)(a+b)}\right)=\frac{a}{a-b}\\a\neq+b, a\neq-b$

8)

$\huge\left[\frac{3a}{8-a^3}\div\frac{4-a^2}{4(a^2+2a+4)}\right]\times \frac{a^2-4a+4}{a}$


$\huge\left[\frac{3a}{8-a^3}\div\frac{4-a^2}{4(a^2+2a+4)}\right]\times \frac{a^2-4a+4}{a}=$ $\huge\frac{3a}{(2-a)(4+2a+a^2)}\times \frac{4(a^2+2a+4)}{(2-a)(2+a)}\times \frac{a^2-4a+4}{a}=$ $\huge\frac{12a}{(2-a)(2-a)(2+a)}\times \frac{(a-2)^2}{a}=\frac{12}{2+a} \\a\neq0, a\neq+2, a\neq-2$

9)

$\huge\left(\frac{4x^2-8x+4}{x^2+1}\div \frac{x+1}{3}\right)\div \frac{6x-6}{x^4-1}$


$\huge\left(\frac{4x^2-8x+4}{x^2+1}\div \frac{x+1}{3}\right)\div \frac{6x-6}{x^4-1}=$ $\huge\frac{(2x-2)^2}{x^2+1}\times \frac{3}{x+1}\times \frac{(x^2-1)(x^2+1)}{6(x-1)}=$ $\huge\frac{(2x-2)^2}{x^2+1}\times \frac{3}{x+1}\times \frac{(x-1)(x+1)(x^2+1)}{6(x-1)}=$ $\huge\frac{3(2x-2)^2}{6}=\frac{(2x-2)^2}{2}=\frac{4x^2-8x+4}{2}=\frac{2(2x^2-4x+2}{2}=2x^2-4x+2=2(x-1)^2 \\ x\neq+1, x\neq-1$

10)

$\huge 6a+\left(\frac{a}{a-2}-\frac{a}{a+2}\right)\div \frac{4a}{a^4-2a^3+8a-16}$


$\huge 6a+\frac{a(a+2)-a(a-2)}{(a-2)(a+2)}\div \frac{4a}{a^3(a-2)8(a-2)}=$ $\huge 6a+\frac{a^2+2a-a^2+2a}{(a-2)(a+2)}\times \frac{a^3(a-2)8(a-2)}{4a}=$ $\huge 6a+\frac{4a}{(a-2)(a+2)}\times \frac{(a-2)(a^3+8)}{4a}=$ $\huge 6a+\frac{(a-2)(a+2)(a^2-2a+4)}{(a-2)(a+2)}=6a+a^2-2a+4=a^2+4a+4=(a+2)^2\\a\neq0, a\neq+2, a\neq-2$

skolaposkole1