Рационал алгебралық өрнектерді тепе-тең түрлендіру (Рустюмова 1.2.1)

Шешуі: $${x^3} + {x^2}y + x{y^2} + {y^3} = \left( {{x^3} + {x^2}y} \right) + \left( {x{y^2} + {y^3}} \right) = $$ $$ = {x^2}(x + y) + {y^2}(x + y) = \left( {{x^2} + {y^2}} \right)(x + y)$$

Шешуі: $${x^3} — 2{x^2}y + {y^2}x — 2{y^3} = \left( {{x^3} + {y^2}x} \right) — \left( {2{x^2}y + 2{y^3}} \right) = $$ $$ = x\left( {{x^2} + {y^2}} \right) — 2y\left( {{x^2} + {y^2}} \right) = \left( {{x^2} + {y^2}} \right)(x — 2y)$$

Шешуі: $${x^4}{y^2} + 2{x^4} + {x^2}{y^2} + 2{x^2} = \left( {{x^4}{y^2} + {x^2}{y^2}} \right) + \left( {2{x^4} + 2{x^2}} \right) = $$ $$ = {x^2}{y^2}\left( {{x^2} + 1} \right) + 2{x^2}\left( {{x^2} + 1} \right) = \left( {{x^2}{y^2} + 2{x^2}} \right)\left( {{x^2} + 1} \right) = $$ $$ = {x^2}\left( {{y^2} + 2} \right)\left( {{x^2} + 1} \right)$$

Шешуі: $${2{x^2} + 7x — 4 = 2(x + 4)\left( {x — \dfrac{1}{2}} \right) = (x + 4)(2x — 1)}$$ $${2{x^2} + 7x — 4 = 0}$$ $${D = 49 + 32 = 81 = {9^2}}$$ $${x_1} = \dfrac{{ — 7 + 9}}{4} = \dfrac{1}{2};$$ $${x_2} = \dfrac{{ — 7 — 9}}{4} = \dfrac{{ — 16}}{4} = — 4.$$

Шешуі: $$4{x^2} + y — 2x — {y^2} = \left( {4{x^2} — {y^2}} \right) + (y — 2x) = $$ $$ = (2x — y)(2x + y) — (2x — y) = (2x — y)(2x + y — 1)$$

Шешуі: $${x^2} — 9x + 9y — {y^2} = \left( {{x^2} — {y^2}} \right) — (9x — 9y) = $$ $$ = (x — y)(x + y) — 9(x — y) = (x — y)(x + y — 9)$$

Шешуі: $$3ax — 2 — x + 6a = (3ax + 6a) — (2 + x) = $$ $$ = 3a(x + 2) — (x + 2) = (x + 2)(3a — 1)$$

Шешуі: $$4xy — 3 — 2y + 6x = (4xy — 2y) + (6x — 3) = $$ $$ = 2y(2x — 1) + 3(2x — 1) = (2x — 1)(2y + 3)$$

Шешуі: $$4{x^2} + 5x — 9{x^2} + 15x = \left( {4{x^2} — 9{x^2}} \right) + (5x + 15x) = $$ $$ = — 5{x^2} + 20x = 5x(4 — x)$$

Шешуі: $${2{x^2} — 7x + 5 = 2\left( {x — \dfrac{5}{2}} \right)(x — 1) = (2x — 5)(x — 1)}$$ $${D = 49 — 40 = 9 = {3^2}}$$ $${x_1} = \dfrac{{7 + 3}}{4} = \dfrac{{10}}{4} = \dfrac{5}{2};$$ $${x_2} = \dfrac{{7 — 3}}{4} = \dfrac{4}{4} = 1.$$

Шешуі: $${6{x^2} — 11x — 30 = 6\left( {x — \dfrac{{10}}{3}} \right)\left( {x + \dfrac{3}{2}} \right) = (3x — 10)(2x + 3)}$$ $${D = 121 + 720 = 841 = {{29}^2}}$$ $${x_1} = \dfrac{{11 + 29}}{{12}} = \dfrac{{40}}{{12}} = \dfrac{{10}}{3};$$ $${x_2} = \dfrac{{11 — 29}}{{12}} = — \dfrac{{18}}{{12}} = — \dfrac{3}{2}.$$

Шешуі: $$4{x^2} — 4{y^2} + 3x + 3y = \left( {4{x^2} — 4{y^2}} \right) + (3x + 3y) = $$ $$ = 4\left( {{x^2} — {y^2}} \right) + 3(x + y) = 4(x — y)(x + y) + 3(x + y) = $$ $$ = (x + y)(4(x — y) + 3) = (x + y)(4x — 4y + 3)$$

Шешуі: $${10{x^2} — 3x — 4 = 10\left( {x — \dfrac{4}{5}} \right)\left( {x + \dfrac{1}{2}} \right) = (5x — 4)(2x + 1)}$$ $${D = 9 + 160 = 169 = {{13}^2}}$$ $${x_1} = \frac{{3 + 13}}{{20}} = \frac{{16}}{{20}} = \frac{4}{5};$$ $${x_2} = \frac{{3 — 13}}{{20}} = — \frac{1}{2}.$$

Шешуі: $${2x + y + {y^2} — 4{x^2} = (2x + y) + \left( {{y^2} — 4{x^2}} \right) = }$$ $${ = (2x + y) + (y — 2x)(y + 2x) = (2x + y)(1 + y — 2x)}$$

Шешуі: $$5{x^2}y — 4x{y^2} — {y^3} = y\left( {5{x^2} — 4xy — {y^2}} \right) = $$ $$ = y\left( {4{x^2} — 4xy + {x^2} — {y^2}} \right) = y\left( {4x(x — y) + (x — y)(x + y)} \right) = $$ $$ = y(x — y)(4x + x + y) = y(x — y)(5x + y)$$

Шешуі: $${x^2}y — 4x{y^2} — 5{y^3} = y\left( {{x^2} — 4xy — 5{y^2}} \right) = $$ $$ = y\left( {{x^2} — {y^2} — 4xy — 4{y^2}} \right) = y\left( {(x — y)(x + y) — 4y(x + y)} \right) = $$ $$ = y(x + y)(x — y — 4y) = y(x + y)(x — 5y)$$

Шешуі: $${a^3} + 6{a^2} + 12a + 8 = {a^3} + 3{a^2} \cdot 2 + 3 \cdot a \cdot {2^2} + {2^3} = {(a + 2)^3}$$

Шешуі: $${4{a^2} — 12ab + 5{b^2} = 4{a^2} — 2ab — 10ab + 5{b^2} = }$$ $${ = 2a(2a — b) — 5b(2a — b) = (2a — b)(2a — 5b)}$$

Шешуі: $${{b^6} + {b^4}{c^2} — {b^2} — {c^2} = {b^4}\left( {{b^2} + {c^2}} \right) — \left( {{b^2} + {c^2}} \right) = }$$ $${ = \left( {{b^2} + {c^2}} \right)\left( {{b^4} — 1} \right) = \left( {{b^2} + {c^2}} \right)\left( {{b^2} + 1} \right)\left( {{b^2} — 1} \right) = \left( {{b^2} + {c^2}} \right)\left( {{b^2} + 1} \right)(b + 1)(b — 1)}$$

Шешуі: $$9{x^2} + 6xy + {y^2} — {z^2} = {(3x + y)^2} — {z^2} = $$ $$ = (3x + y — z)(3x + y + z)$$

Шешуі: $${(x + 1)^3} — 3{(x + 1)^2} + 3(x + 1) — 1 = {(x + 1 — 1)^3} = {x^3}$$

Шешуі: $$a — 3b + 9{b^2} — {a^2} = (a — 3b) + \left( {9{b^2} — {a^2}} \right) = $$ $$ = (a — 3b) — \left( {{a^2} — 9{b^2}} \right) = (a — 3b) — (a — 3b)(a + 3b) = $$ $$ = (a — 3b)(1 — a — 3b)$$

Шешуі: $${{x^2} + {y^2} + 4 + 2xy + 4x + 4y = \left( {{x^2} + 2xy + {y^2}} \right) + (4x + 4y) + 4 = }$$ $${ = {{(x + y)}^2} + 4(x + y) + 4 = {{(x + y + 2)}^2}}$$

Шешуі: $$2{a^4} + 30{a^3} + 150{a^2} + 250a = $$ $$ = 2a\left( {{a^3} + 15{a^2} + 75a + 125} \right) = 2a{(a + 5)^3}$$

Шешуі: $$64{(2 — 5a)^2} — 25{(6a — 5)^2} = $$ $$ = \left( {8(2 — 5a) + 5(6a — 5)} \right)\left( {8(2 — 5a) — 5(6a — 5)} \right) = $$ $$ = (16 — 40a + 30a — 25)(16 — 40a — 30a + 25) = $$ $$ = ( — 9 — 10a)(41 — 70a) = — (10a + 9) \cdot ( — (70a — 41)) = $$ $$ = (10a + 9)(70a — 41)$$