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
$$27{c^3} – 3{c^2} + 2c – 8 = \left( {27{c^3} – 8} \right) – \left( {3{c^2} – 2c} \right) = $$ $$ = (3c – 2)\left( {9{c^2} + 6c + 4} \right) – c(3c – 2) = (3c – 2)\left( {9{c^2} + 5c + 4} \right)$$