Бөлшектерді қысқартыңыз.
№ 1 Өрнекті ықшамдаңыз: $\dfrac{{{x^2} – x – 2}}{{3{x^2} – 5x – 2}}$
Шешуі: $${3{x^2} – 5x – 2 = 0}$$ $${D = 25 + 24 = 49 = 7^2}$$ $${x_1} = \dfrac{{5 + 7}}{6} = \dfrac{{12}}{6} = 2,$$ $${x_2} = \dfrac{{5 – 7}}{6} = – \dfrac{1}{3}.$$ $$\boxed{a{x^2} + bx + c = a\left( {x – {x_1}} \right)\left( {x – {x_2}} \right)}$$ $$3{x^2} – 5x – 2 = 3\left( {x + \dfrac{1}{3}} \right)(x – 2)$$ $${x^2} – x – 2 = {x^2} + x – 2x – 2 = x\left( {x + 1} \right) – 2\left( {x + 1} \right) = \left( {x + 1} \right)\left( {x – 2} \right)$$ $$\dfrac{{{x^2} – x – 2}}{{3{x^2} – 5x – 2}} = \dfrac{{(x – 2)(x + 1)}}{{3\left( {x + \dfrac{1}{3}} \right)(x – 2)}} = \dfrac{{(x – 2)(x + 1)}}{{(3x + 1)(x – 2)}} = \dfrac{{x + 1}}{{3x + 1}}$$
№ 2 Өрнекті ықшамдаңыз: $\dfrac{{{x^3} + 4{x^2} – 9x – 36}}{{{x^2} + x – 12}}$
Шешуі: $$ = \dfrac{{\left( {{x^3} + 4{x^2}} \right) – 9(x + 4)}}{{(x + 4)(x – 3)}} = \dfrac{{{x^2}(x + 4) – 9(x + 4)}}{{(x + 4)(x – 3)}} = $$ $$ = \dfrac{{(x + 4)\left( {{x^2} – 9} \right)}}{{(x + 4)(x – 3)}} = \dfrac{{(x – 3)(x + 3)}}{{(x – 3)}} = x + 3$$
№ 3 Өрнекті ықшамдаңыз: $\dfrac{{5{x^2} – 7x – 5xy + 7y}}{{x – y}}$
Шешуі: $$ = \dfrac{{\left( {5{x^2} – 5xy} \right) – 7(x – y)}}{{x – y}} = \dfrac{{5x(x – y) – 7(x – y)}}{{x – y}} = $$ $$ = \dfrac{{(x – y)(5x – 7)}}{{x – y}} = 5x – 7$$
№ 4 Өрнекті ықшамдаңыз: $\dfrac{{{a^4} – 1}}{{1 – {a^8}}}$
Шешуі: $$ = \dfrac{{ – \left( {1 – {a^4}} \right)}}{{\left( {1 – {a^4}} \right)\left( {1 + {a^4}} \right)}} = – \dfrac{1}{{1 + {a^4}}}$$
№ 5 Өрнекті ықшамдаңыз: $\dfrac{{{x^4} + {a^3}x}}{{{x^3} – a{x^2} + {a^2}x}}$
Шешуі: $$ = \dfrac{{x\left( {{x^3} + {a^3}} \right)}}{{x\left( {{x^2} – ax + {a^2}} \right)}} = \dfrac{{(x + a)\left( {{x^2} – ax + {a^2}} \right)}}{{{x^2} – ax + {a^2}}} = x + a$$
№ 6 Өрнекті ықшамдаңыз: $\dfrac{{1 + ab – a – b}}{{1 – 2a + {a^2}}}$
Шешуі: $$ = \dfrac{{(1 – b) + (ab – a)}}{{{{(1 – a)}^2}}} = \dfrac{{(1 – b) – a(1 – b)}}{{{{(1 – a)}^2}}} = \dfrac{{(1 – b)(1 – a)}}{{{{(1 – a)}^2}}} = \dfrac{{1 – b}}{{1 – a}}$$
№ 7 Өрнекті ықшамдаңыз: $$\dfrac{{xy – y + 2x – 2}}{{xy + 3x – y – 3}}$$
Шешуі: $$ = \dfrac{{y(x – 1) + 2(x – 1)}}{{x(y + 3) – (y + 3)}} = \dfrac{{(x – 1)(y + 2)}}{{(y + 3)(x – 1)}} = \dfrac{{y + 2}}{{y + 3}}$$
№ 8 Өрнекті ықшамдаңыз: $\dfrac{{{x^2} – 9x + 14}}{{{x^2} – 10x + 16}}$
Шешуі: $$ = \dfrac{{(x – 7)(x – 2)}}{{(x – 8)(x – 2)}} = \dfrac{{x – 7}}{{x – 8}}$$
№ 9 Өрнекті ықшамдаңыз: $\dfrac{{7{x^2}{y^4} + 7{x^4}{y^2}}}{{{x^6} + {y^6}}}$
Шешуі: $$ = \dfrac{{7{x^2}{y^2}\left( {{y^2} + {x^2}} \right)}}{{\left( {{x^2} + {y^2}} \right)\left( {{x^4} – {x^2}{y^2} + {y^4}} \right)}} = \dfrac{{7{x^2}{y^2}}}{{{x^4} – {x^2}{y^2} + {y^4}}}$$
№ 10 Өрнекті ықшамдаңыз: $\dfrac{{{x^4} + {x^2} + 1}}{{{x^2} + x + 1}}$
Шешуі: 1-жолы: $${x^4} + {x^2} + 1 = \left( {{x^4} + 2{x^2} + 1} \right) – 2{x^2} + {x^2} = $$ $$ = {\left( {{x^2} + 1} \right)^2} – {x^2} = \left( {{x^2} + 1 – x} \right)\left( {{x^2} + 1 + x} \right) = \left( {{x^2} – x + 1} \right)\left( {{x^2} + x + 1} \right)$$ $$\dfrac{{{x^4} + {x^2} + 1}}{{{x^2} + x + 1}} = \dfrac{{\left( {{x^2} + x + 1} \right)\left( {{x^2} – x + 1} \right)}}{{{x^2} + x + 1}} = {x^2} – x + 1$$ 2-жолы: 
№ 11 Өрнекті ықшамдаңыз: ${\dfrac{{{a^{36}} – {a^3}}}{{{a^{26}} + {a^{15}} + {a^4}}}}$
Шешуі: $${ = \dfrac{{{a^3}\left( {{a^{33}} – 1} \right)}}{{{a^4}\left( {{a^{22}} + {a^{11}} + 1} \right)}} = \dfrac{{\left( {{a^{11}} – 1} \right)\left( {{a^{22}} + {a^{11}} + 1} \right)}}{{a\left( {{a^{22}} + {a^{11}} + 1} \right)}} = \dfrac{{{a^{11}} – 1}}{a}}$$
№ 12 Өрнекті ықшамдаңыз: ${\dfrac{{{a^2} + {b^2} – {c^2} – 2ab}}{{{a^2} – {b^2} + {c^2} + 2ac}}}$
Шешуі: $${ = \dfrac{{\left( {{a^2} + {b^2} – 2ab} \right) – {c^2}}}{{\left( {{a^2} + {c^2} + 2ac} \right) – {b^2}}} = \dfrac{{{{(a – b)}^2} – {c^2}}}{{{{(a + c)}^2} – {b^2}}} = }$$ $${ = \dfrac{{(a – b – c)(a – b + c)}}{{(a + c – b)(a + c + b)}} = \dfrac{{a – b – c}}{{a + b + c}}}$$
№ 13 Өрнекті ықшамдаңыз: $\dfrac{{{x^6} – 1}}{{3{x^3} – 3x}}$
Шешуі: $$ = \dfrac{{\left( {{x^2} – 1} \right)\left( {{x^4} + {x^2} + 1} \right)}}{{3x\left( {{x^2} – 1} \right)}} = \dfrac{{{x^4} + {x^2} + 1}}{{3x}}$$
№ 14 Өрнекті ықшамдаңыз: $\dfrac{{{m^2} – {n^2} – 2np – {p^2}}}{{{m^2} – 2mn + {n^2} – {p^2}}}$
Шешуі: $$ = \dfrac{{{m^2} – \left( {{n^2} + 2np + {p^2}} \right)}}{{\left( {{m^2} – 2mn + {n^2}} \right) – {p^2}}} = \dfrac{{{m^2} – {{(n + p)}^2}}}{{{{(m – n)}^2} – {p^2}}} = $$ $${ = \dfrac{{(m – n – p)(m + n + p)}}{{(m – n – p)(m – n + p)}} = \dfrac{{m + n + p}}{{m – n + p}}}$$
№ 15 Өрнекті ықшамдаңыз: ${\dfrac{{{a^3} – 2{a^2} + 4a – 3}}{{{a^2} – 7a + 6}}}$
Шешуі: $${{a^3} – 2{a^2} + 4a – 3 = \left( {{a^3} – 2{a^2} + a} \right) + (3a – 3) = }$$ $${ = a\left( {{a^2} – 2a + 1} \right) + 3(a – 1) = a{{(a – 1)}^2} + 3(a – 1) = }$$ $${ = (a – 1)(a(a – 1) + 3) = (a – 1)\left( {{a^2} – a + 3} \right)}$$ $$\dfrac{{{a^3} – 2{a^2} + 4a – 3}}{{{a^2} – 7a + 6}} = \dfrac{{(a – 1)\left( {{a^2} – a + 3} \right)}}{{(a – 1)(a – 6)}} = \dfrac{{{a^2} – a + 3}}{{a – 6}}$$
№ 16 Өрнекті ықшамдаңыз: ${\dfrac{{x + y + {x^2} – {y^2}}}{{x – y + {x^2} – 2xy + {y^2}}}}$
Шешуі: $$ = \dfrac{{(x + y) + (x – y)(x + y)}}{{(x – y) + {{(x – y)}^2}}} = \dfrac{{(x + y)(1 + x – y)}}{{(x – y)(1 + x – y)}} = \dfrac{{x + y}}{{x – y}}$$
№ 17 Өрнекті ықшамдаңыз: ${\dfrac{{ab + b + az + z}}{{{a^3} + 3{a^2} + 3a + 1}}}$
Шешуі: $$ = \dfrac{{b(a + 1) + z(a + 1)}}{{{{(a + 1)}^3}}} = \dfrac{{(a + 1)(b + z)}}{{{{(a + 1)}^3}}} = \dfrac{{b + z}}{{{{(a + 1)}^2}}}$$
№ 18 Өрнекті ықшамдаңыз: $\dfrac{{{x^3} + 2{x^2} + 4x + 3}}{{{x^2} + 7x + 6}}$
Шешуі:$${x^3} + 2{x^2} + 4x + 3 = \left( {{x^3} + 2{x^2} + x} \right) + (3x + 3) = $$ $$ = x\left( {{x^2} + 2x + 1} \right) + 3(x + 1) = x{(x + 1)^2} + 3(x + 1) = $$ $$ = (x + 1)(x(x + 1) + 3) = (x + 1)\left( {{x^2} + x + 3} \right)$$ $$\dfrac{{{x^3} + 2{x^2} + 4x + 3}}{{{x^2} + 7x + 6}} = \dfrac{{(x + 1)\left( {{x^2} + x + 3} \right)}}{{(x + 1)(x + 6)}} = \dfrac{{{x^2} + x + 3}}{{x + 6}}$$
№ 19 Өрнекті ықшамдаңыз: ${\dfrac{{{x^4} + {x^2}{y^2} + {y^4}}}{{{x^2} – xy + {y^2}}}}$
Шешуі: $${{x^4} + {x^2}{y^2} + {y^4} = \left( {{x^4} + 2{x^2}{y^2} + {y^4}} \right) – 2{x^2}{y^2} + {x^2}{y^2} = }$$ $${ = {{\left( {{x^2} + {y^2}} \right)}^2} – {x^2}{y^2} = \left( {{x^2} + {y^2} – xy} \right)\left( {{x^2} + {y^2} + xy} \right)}$$ $${ = \left( {{x^2} – xy + {y^2}} \right)\left( {{x^2} + xy + {y^2}} \right)}$$ $${\dfrac{{{x^4} + {x^2}{y^2} + {y^4}}}{{{x^2} – xy + {y^2}}} = \dfrac{{\left( {{x^2} – xy + {y^2}} \right)\left( {{x^2} + xy + {y^2}} \right)}}{{{x^2} – xy + {y^2}}} = {x^2} + xy + {y^2}}$$
№ 20 Өрнекті ықшамдаңыз: ${\dfrac{{{x^4} + {x^2}{y^2} + {y^4}}}{{(x – y)\left( {{x^3} – {y^3}} \right)}}}$
Шешуі: $${ = \dfrac{{\left( {{x^2} – xy + {y^2}} \right)\left( {{x^2} + xy + {y^2}} \right)}}{{(x – y)(x – y)\left( {{x^2} + xy + {y^2}} \right)}} = \dfrac{{{x^2} – xy + {y^2}}}{{{{(x – y)}^2}}}}$$
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