Қысқаша көбейту формулаларын пайдаланып, көпмүшелерді көбейткіштерге жіктеу
2.2. Көпмүшені көбейткіштерге жіктеңіз.:
1) $x^6-1$
Шешуі
$${x^6} — 1 = {\left( {{x^3}} \right)^2} — 1 = \left( {{x^3} — 1} \right)\left( {{x^3} + 1} \right) = $$ $$ = (x — 1)(x + 1)\left( {{x^2} + x + 1} \right)\left( {{x^2} — x + 1} \right)$$
2) $x^3-x+2 x+2$
Шешуі
$${x^3} — x + 2x + 2 = \left( {{x^3} — x} \right) + (2x + 2) = x(x — 1)(x + 1) + 2(x + 1) = $$ $$ = (x + 1)(x(x — 1) + 2) = (x + 1)\left( {{x^2} — x + 2} \right)$$
3) $y^3-3 y^2+6 y-8$
Шешуі
$${y^3} — 3{y^2} + 6y — 8 = \left( {{y^3} — 8} \right) + \left( {6y — 3{y^2}} \right) = $$ $$ = (y — 2)\left( {{y^2} + 2y + 4} \right) — 3y(y — 2) = (y — 2)\left( {{y^2} — y + 4} \right)$$
4) $a^4-2 a^3+a^2-1$
Шешуі
$${a^4} — 2{a^3} + {a^2} — 1 = {a^2}\left( {{a^2} — 2a + 1} \right) — 1 = $$ $$ = {a^2}{(a — 1)^2} — 1 = (a(a — 1) — 1)(a(a — 1) + 1) = $$ $$ = \left( {{a^2} — a — 1} \right)\left( {{a^2} — a + 1} \right)$$
5) $a^2-2 a b+b^2-z^2$
Шешуі
$${a^2} — 2ab + {b^2} — {z^2} = {(a — b)^2} — {z^2} = (a — b — z)(a — b + z)$$
6) $a-3 b+9 b^2-a^2$
Шешуі
$$a — 3b + 9{b^2} — {a^2} = (a — 3b) — (a — 3b)(a + 3b) = (a — 3b)(1 — a — 3b)$$
7) $64(2-5 a)^2-25(6 a-5)^2$
Шешуі
$$64{(2 — 5a)^2} — 25{(6a — 5)^2} = {\left( {8(2 — 5a)} \right)^2} — {\left( {5(6a — 5)} \right)^2} = $$ $$ = \left( {8(2 — 5a) + 5(6a — 5)} \right)\left( {8(2 — 5a) — 5(6a — 5)} \right) = $$ $$ = ( — 9 — 10a)( — 70a + 41) = (10a + 9)(70a — 41)$$
8) $a^2 b+b^2 c+a c^2-a b^2-b c^2-a^2 c$
Шешуі
$${a^2}b + {b^2}c + a{c^2} — a{b^2} — h{c^2} — {a^2}c = $$ $$ = \left( {{a^2}b — {a^2}c} \right) + \left( {{b^2}c — b{c^2}} \right) + \left( {a{c^2} — a{b^2}} \right) = $$ $$ = {a^2}(b — c) + bc(b — c) — a(b — c)(b + c) = $$ $$ = (b — c)\left( {{a^2} + bc — ab — ac} \right) = $$ $$ = (b — c)\left( {a(a — c) — b(a — c)} \right) = (b — c)(a — c)(a — b)$$
9) $4 x^2-4 x^3+12 x^2 y-9 y^2-9 x y^2$
Шешуі
$$4{x^2} — 4{x^3} + 12{x^2}y — 9{y^2} — 9x{y^2} = \left( {4{x^2} — 9{y^2}} \right) — \left( {4{x^3} — 12{x^2}y + 9x{y^2}} \right) = $$ $$ = (2x — 3y)(2x + 3y) — x{(2x — 3y)^2} = (2x — 3y)\left( {2x + 3y — 2{x^2} + 3xy} \right)$$
10) $27 c^3-3 c^2+2 c-8$
Шешуі
$$27{c^3} — 3{c^2} + 2c — 8 = \left( {27{c^3} — 8} \right) — \left( {3{c^2} — 2c} \right) = $$ $$ = (3c — 2)\left( {9{c^2} + 6c + 4} \right) — c(3c — 2) = (3c — 2)\left( {9{c^2} + 5c + 4} \right)$$