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а)
![frac{2a + b}{2 {a}^{2} - ab } - frac{16a}{4 {a}^{2} - {b}^{2} } - frac{2a - b}{2 {a}^{2} + ab} = frac{2a + b}{a(2a - b)} - frac{16a}{(2a - b)(2a + b)} - frac{2a - b}{a(2a + b)} = frac{ {(2a + b)}^{2} - 16 {a}^{2} - {(2a - b)}^{2} }{a(2a - b)(2a + b)} = frac{(2a + b + 2a - b)(2a + b - 2a + b) - 16 {a}^{2} }{a(2a - b)(2a + b)} = frac{4a times 2b - 16 {a}^{2} }{a(2a - b)(2a + b)} = frac{8a(b - 2a)}{a(2a - b)(2a + b)} = - frac{8}{2a + b} frac{2a + b}{2 {a}^{2} - ab } - frac{16a}{4 {a}^{2} - {b}^{2} } - frac{2a - b}{2 {a}^{2} + ab} = frac{2a + b}{a(2a - b)} - frac{16a}{(2a - b)(2a + b)} - frac{2a - b}{a(2a + b)} = frac{ {(2a + b)}^{2} - 16 {a}^{2} - {(2a - b)}^{2} }{a(2a - b)(2a + b)} = frac{(2a + b + 2a - b)(2a + b - 2a + b) - 16 {a}^{2} }{a(2a - b)(2a + b)} = frac{4a times 2b - 16 {a}^{2} }{a(2a - b)(2a + b)} = frac{8a(b - 2a)}{a(2a - b)(2a + b)} = - frac{8}{2a + b}](https://tex.z-dn.net/?f=+frac%7B2a+%2B+b%7D%7B2+%7Ba%7D%5E%7B2%7D+-+ab+%7D++-++frac%7B16a%7D%7B4+%7Ba%7D%5E%7B2%7D+-++%7Bb%7D%5E%7B2%7D++%7D++-++frac%7B2a+-+b%7D%7B2+%7Ba%7D%5E%7B2%7D++%2B+ab%7D+%3D++frac%7B2a+%2B+b%7D%7Ba%282a+-+b%29%7D+++-++frac%7B16a%7D%7B%282a+-+b%29%282a+%2B+b%29%7D++-++frac%7B2a+-+b%7D%7Ba%282a+%2B+b%29%7D++%3D++frac%7B+%7B%282a+%2B+b%29%7D%5E%7B2%7D++-+16+%7Ba%7D%5E%7B2%7D++-++%7B%282a+-+b%29%7D%5E%7B2%7D+%7D%7Ba%282a+-+b%29%282a+%2B+b%29%7D++%3D++frac%7B%282a+%2B+b+%2B+2a+-+b%29%282a+%2B+b+-+2a+%2B+b%29+-+16+%7Ba%7D%5E%7B2%7D+%7D%7Ba%282a+-+b%29%282a+%2B+b%29%7D++%3D++frac%7B4a+times+2b+-+16+%7Ba%7D%5E%7B2%7D+%7D%7Ba%282a+-+b%29%282a+%2B+b%29%7D++%3D++frac%7B8a%28b+-+2a%29%7D%7Ba%282a+-+b%29%282a+%2B+b%29%7D++%3D++-++frac%7B8%7D%7B2a+%2B+b%7D+)
б)
![frac{1}{ {(a - 3)}^{2} } - frac{2}{ {a}^{2} - 9} + frac{1}{ {(a + 3)}^{2} } = frac{ {(a + 3)}^{2} - 2(a - 3)(a + 3) + {(a - 3)}^{2} }{ {(a - 3)}^{2} {(a + 3)}^{2} } = frac{ {(a + 3 - a + 3)}^{2} }{ {(a + 3)}^{2} {(a - 3)}^{2} } = frac{36}{ {(a + 3)}^{2} {(a - 3)}^{2} } frac{1}{ {(a - 3)}^{2} } - frac{2}{ {a}^{2} - 9} + frac{1}{ {(a + 3)}^{2} } = frac{ {(a + 3)}^{2} - 2(a - 3)(a + 3) + {(a - 3)}^{2} }{ {(a - 3)}^{2} {(a + 3)}^{2} } = frac{ {(a + 3 - a + 3)}^{2} }{ {(a + 3)}^{2} {(a - 3)}^{2} } = frac{36}{ {(a + 3)}^{2} {(a - 3)}^{2} }](https://tex.z-dn.net/?f=+frac%7B1%7D%7B+%7B%28a+-+3%29%7D%5E%7B2%7D+%7D++-++frac%7B2%7D%7B+%7Ba%7D%5E%7B2%7D++-+9%7D++%2B++frac%7B1%7D%7B+%7B%28a+%2B+3%29%7D%5E%7B2%7D+%7D++%3D++frac%7B++%7B%28a+%2B+3%29%7D%5E%7B2%7D+-+2%28a+-+3%29%28a+%2B+3%29+++%2B++%7B%28a+-+3%29%7D%5E%7B2%7D+%7D%7B+%7B%28a+-+3%29%7D%5E%7B2%7D++%7B%28a+%2B+3%29%7D%5E%7B2%7D+%7D++%3D++frac%7B+%7B%28a+%2B+3+-+a+%2B+3%29%7D%5E%7B2%7D+%7D%7B+%7B%28a+%2B+3%29%7D%5E%7B2%7D+%7B%28a+-+3%29%7D%5E%7B2%7D++%7D++%3D++frac%7B36%7D%7B+%7B%28a+%2B+3%29%7D%5E%7B2%7D++%7B%28a+-+3%29%7D%5E%7B2%7D+%7D+)
в)
![frac{x - 2}{ {x}^{2} + 2x + 4} - frac{6x}{ {x}^{3} - 8} + frac{1}{x - 2} = frac{x - 2}{ {x}^{2} + 2x + 4 } - frac{6x}{(x - 2)( {x}^{2} + 2x + 4)} + frac{1}{x - 2} = frac{ {(x - 2)}^{2} - 6x + {x}^{2} + 2x + 4}{(x - 2)( {x}^{2} + 2x + 4) } = frac{ {x}^{2} - 4x + 4 - 6x + {x}^{2} + 2x + 4 }{(x - 2)( {x}^{2} + 2x + 4) } = frac{2 {x}^{2} - 8x + 8}{(x - 2)( {x}^{2} + 2x + 4)} = frac{2( {x}^{2} - 4x + 4) }{(x - 2)( {x}^{2} + 2x + 4) } = frac{2 {(x - 2)}^{2} }{ (x - 2)( {x}^{2} + 2x + 4) } = frac{2(x - 2)}{ {x}^{2} + 2x + 4} frac{x - 2}{ {x}^{2} + 2x + 4} - frac{6x}{ {x}^{3} - 8} + frac{1}{x - 2} = frac{x - 2}{ {x}^{2} + 2x + 4 } - frac{6x}{(x - 2)( {x}^{2} + 2x + 4)} + frac{1}{x - 2} = frac{ {(x - 2)}^{2} - 6x + {x}^{2} + 2x + 4}{(x - 2)( {x}^{2} + 2x + 4) } = frac{ {x}^{2} - 4x + 4 - 6x + {x}^{2} + 2x + 4 }{(x - 2)( {x}^{2} + 2x + 4) } = frac{2 {x}^{2} - 8x + 8}{(x - 2)( {x}^{2} + 2x + 4)} = frac{2( {x}^{2} - 4x + 4) }{(x - 2)( {x}^{2} + 2x + 4) } = frac{2 {(x - 2)}^{2} }{ (x - 2)( {x}^{2} + 2x + 4) } = frac{2(x - 2)}{ {x}^{2} + 2x + 4}](https://tex.z-dn.net/?f=+frac%7Bx+-+2%7D%7B+%7Bx%7D%5E%7B2%7D++%2B+2x+%2B+4%7D++-+frac%7B6x%7D%7B+%7Bx%7D%5E%7B3%7D++-+8%7D++%2B++frac%7B1%7D%7Bx+-+2%7D++%3D++frac%7Bx+-+2%7D%7B+%7Bx%7D%5E%7B2%7D+%2B+2x+%2B+4+%7D++-++frac%7B6x%7D%7B%28x+-+2%29%28+%7Bx%7D%5E%7B2%7D++%2B+2x+%2B+4%29%7D++%2B++frac%7B1%7D%7Bx+-+2%7D++%3D++frac%7B+%7B%28x+-+2%29%7D%5E%7B2%7D++-+6x+%2B++%7Bx%7D%5E%7B2%7D++%2B+2x+%2B+4%7D%7B%28x+-+2%29%28+%7Bx%7D%5E%7B2%7D+%2B+2x+%2B+4%29+%7D++%3D++frac%7B+%7Bx%7D%5E%7B2%7D+-+4x+%2B+4+-+6x+%2B++%7Bx%7D%5E%7B2%7D++%2B+2x+%2B+4+%7D%7B%28x+-+2%29%28+%7Bx%7D%5E%7B2%7D+%2B+2x+%2B+4%29+%7D++%3D++frac%7B2+%7Bx%7D%5E%7B2%7D++-+8x+%2B+8%7D%7B%28x+-+2%29%28+%7Bx%7D%5E%7B2%7D++%2B+2x+%2B+4%29%7D++%3D++frac%7B2%28+%7Bx%7D%5E%7B2%7D+-+4x+%2B+4%29+%7D%7B%28x+-+2%29%28+%7Bx%7D%5E%7B2%7D+%2B+2x+%2B+4%29+%7D++%3D++frac%7B2+%7B%28x+-+2%29%7D%5E%7B2%7D+%7D%7B+%28x+-+2%29%28+%7Bx%7D%5E%7B2%7D+%2B+2x+%2B+4%29++%7D++%3D++frac%7B2%28x+-+2%29%7D%7B+%7Bx%7D%5E%7B2%7D++%2B+2x+%2B+4%7D+)
г)
![frac{2 {a}^{2} + 7a + 3 }{ {a}^{3} - 1 } - frac{1 - 2a}{ {a}^{2} + a + 1 } - frac{3}{a - 1} = frac{2 {a}^{2} + 7a + 3 }{(a - 1)( {a}^{2} + a + 1)} - frac{(1 - 2a)(a - 1)}{ ({a}^{2} + a + 1)(a - 1) } - frac{3( {a}^{2} + a + 1) }{(a - 1)( {a}^{2} + a + 1)} = frac{2 {a}^{2} + 7a + 3 - a + 2 {a}^{2} + 1 - 2a - 3 {a}^{2} - 3a - 3 }{(a - 1)( {a}^{2} + a + 1) } = frac{ {a}^{2} + a + 1}{(a - 1)( {a}^{2} + a + 1) } = frac{1}{a - 1} frac{2 {a}^{2} + 7a + 3 }{ {a}^{3} - 1 } - frac{1 - 2a}{ {a}^{2} + a + 1 } - frac{3}{a - 1} = frac{2 {a}^{2} + 7a + 3 }{(a - 1)( {a}^{2} + a + 1)} - frac{(1 - 2a)(a - 1)}{ ({a}^{2} + a + 1)(a - 1) } - frac{3( {a}^{2} + a + 1) }{(a - 1)( {a}^{2} + a + 1)} = frac{2 {a}^{2} + 7a + 3 - a + 2 {a}^{2} + 1 - 2a - 3 {a}^{2} - 3a - 3 }{(a - 1)( {a}^{2} + a + 1) } = frac{ {a}^{2} + a + 1}{(a - 1)( {a}^{2} + a + 1) } = frac{1}{a - 1}](https://tex.z-dn.net/?f=+frac%7B2+%7Ba%7D%5E%7B2%7D+%2B+7a+%2B+3+%7D%7B+%7Ba%7D%5E%7B3%7D+-+1+%7D++-++frac%7B1+-+2a%7D%7B+%7Ba%7D%5E%7B2%7D+%2B+a++%2B+1+%7D++-++frac%7B3%7D%7Ba+-+1%7D++%3D++frac%7B2+%7Ba%7D%5E%7B2%7D++%2B+7a+%2B+3+%7D%7B%28a+-+1%29%28+%7Ba%7D%5E%7B2%7D+%2B+a+%2B+1%29%7D++-++frac%7B%281+-+2a%29%28a+-+1%29%7D%7B+%28%7Ba%7D%5E%7B2%7D+%2B+a+%2B+1%29%28a+-+1%29+%7D++-++frac%7B3%28+%7Ba%7D%5E%7B2%7D+%2B+a+%2B+1%29+%7D%7B%28a+-+1%29%28+%7Ba%7D%5E%7B2%7D++%2B+a+%2B+1%29%7D++%3D++frac%7B2+%7Ba%7D%5E%7B2%7D+%2B+7a+%2B+3+-+a+%2B+2+%7Ba%7D%5E%7B2%7D+%2B+1+-+2a+-+3+%7Ba%7D%5E%7B2%7D+-+3a+-+3+++%7D%7B%28a+-+1%29%28+%7Ba%7D%5E%7B2%7D+%2B+a+%2B+1%29+%7D++%3D++frac%7B+%7Ba%7D%5E%7B2%7D++%2B+a+%2B+1%7D%7B%28a+-+1%29%28+%7Ba%7D%5E%7B2%7D+%2B+a+%2B+1%29+%7D++%3D++frac%7B1%7D%7Ba+-+1%7D+)
Всё это легко решается, если знать и видеть формулы сокращенного умножения.
б)
в)
г)
Всё это легко решается, если знать и видеть формулы сокращенного умножения.
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