Разложите многочлен на множители: 1) 5x^3 - 45x;

Пример разложения на множители: $5x^3 - 45x = 5x(x^2 - 9) = 5x(x - 3)(x + 3)$. Разложение на множители: 1) $5x^3 - 45x = 5x(x^2 - 9) = 5x(x - 3)(x + 3)$ 2) $8a^3 - 12ab^2 = 4a(2a^2 - 3b^2)$ 3) $-63mn^2 + 28m^3 = 7m(4m^2 - 9n^2) = 7m(2m - 3n)(2m + 3n)$ 4) $18m^2 - 32m^4 = 2m^2(9 - 16m^2) = 2m^2(3 - 4m)(3 + 4m)$ 5) $3x^5 - 3x = 3x(x^4 - 1) = 3x(x^2 - 1)(x^2 + 1) = 3x(x - 1)(x + 1)(x^2 + 1)$ 6) $2a^4b - 2b^5 = 2b(a^4 - b^4) = 2b(a^2 - b^2)(a^2 + b^2) = 2b(a - b)(a + b)(a^2 + b^2)$ 7) $5m^2 - 10mn + 5n^2 = 5(m^2 - 2mn + n^2) = 5(m - n)^2$ 8) $12a^2 + 36ab + 27b^2 = 3(4a^2 + 12ab + 9b^2) = 3(2a + 3b)^2$ 9) $-9x^2 - 6x - 1 = -(9x^2 + 6x + 1) = -(3x + 1)^2$ 10) $2a^3 - 20a^2b + 50ab^2 = 2a(a^2 - 10ab + 25b^2) = 2a(a - 5b)^2$ 11) $15a + 10b - 5ab - 30 = 5(3a + 2b - ab - 6) = 5(3(a - 2) - b(a - 2)) = 5(a - 2)(3 - b)$ 12) $75 + 15n - 12mn - 60m = 15(5 + n) - 12m(n + 5) = (5 + n)(15 - 12m) = 3(5 + n)(5 - 4m)$ 13) $-30k + 30 - 10p + 10kp = -10(3k - 3 + p - kp) = -10(3(k - 1) - p(k - 1)) = -10(k - 1)(3 - p) = 10(k - 1)(p - 3)$ 14) $12x^2 + 12y - 16x - 9xy = 4x(3x - 4) + 3y(4 - 3x) = (3x - 4)(4x - 3y)$ 15) $-15a^2 - 60ab + 48b + 12a = -3(5a^2 + 20ab - 16b - 4a) = -3(5a(a - 4) + 4b(5a - 4))$ (Данное выражение не раскладывается простым группированием) 16) $-24m^2 - 24n - 16m - 36mn = -4(6m^2 + 6n + 4m + 9mn) = -4(2m(3m + 2) + 3n(2 + 3m)) = -4(3m + 2)(2m + 3n)$ 17) $2kp^2 - 4k^2 + 4k^2p - 2kp = 2kp(p - 1) + 4k^2(p - 1) = (p - 1)(2kp + 4k^2) = 2k(p - 1)(p + 2k)$ 18) $m^2 - x^2 - 2xy - y^2 = m^2 - (x^2 + 2xy + y^2) = m^2 - (x + y)^2 = (m - x - y)(m + x + y)$ 19) $a^2 + b^2 - c^2 - 2ab = (a^2 - 2ab + b^2) - c^2 = (a - b)^2 - c^2 = (a - b - c)(a - b + c)$ 20) $9a^2 - b^2 - 24a + 16 = (9a^2 - 24a + 16) - b^2 = (3a - 4)^2 - b^2 = (3a - 4 - b)(3a - 4 + b)$ 21) $x^2 - y^2 + 3x - 3y = (x - y)(x + y) + 3(x - y) = (x - y)(x + y + 3)$ 22) $4a + 4b - a^2 + b^2 = 4(a + b) + (b^2 - a^2) = 4(a + b) + (b - a)(b + a) = (a + b)(4 + b - a)$ 23) $4x^2 - 4y^2 + 3x + 3y = 4(x - y)(x + y) + 3(x + y) = (x + y)(4x - 4y + 3)$ 24) $x^3 - x^2y - x + y = x^2(x - y) - 1(x - y) = (x - y)(x^2 - 1) = (x - y)(x - 1)(x + 1)$ 25) $4a - ab^2 + 4b - b^3 = a(4 - b^2) + b(4 - b^2) = (4 - b^2)(a + b) = (2 - b)(2 + b)(a + b)$ 26) $bc^2 - 2c^2 - b^3 + 2b^2 = c^2(b - 2) - b^2(b - 2) = (b - 2)(c^2 - b^2) = (b - 2)(c - b)(c + b)$ 27) $x^3 + 3x^2 + 3x + 2 = (x^3 + 1) + (3x^2 + 3x + 1) = (x + 1)(x^2 - x + 1) + 3x(x + 1) + 1 = (x + 1)(x^2 + 2x + 1) = (x + 1)^3$ 28) $y^3 - 5y^2 + 5y - 1 = (y^3 - 1) - 5y(y - 1) = (y - 1)(y^2 + y + 1) - 5y(y - 1) = (y - 1)(y^2 - 4y + 1)$ 29) $7a^3 + a^2 + a + 7 = 7(a^3 + 1) + a(a + 1) = 7(a + 1)(a^2 - a + 1) + a(a + 1) = (a + 1)(7a^2 - 7a + 7 + a) = (a + 1)(7a^2 - 6a + 7)$ 30) $8b^3 + 3b^2 - 3b - 8 = (8b^3 - 8) + (3b^2 - 3b) = 8(b - 1)(b^2 + b + 1) + 3b(b - 1) = (b - 1)(8b^2 + 8b + 8 + 3b) = (b - 1)(8b^2 + 11b + 8)$ 31) $x^5 - x^3y^2 - x^2y^3 + y^5 = x^3(x^2 - y^2) - y^3(x^2 - y^2) = (x^2 - y^2)(x^3 - y^3) = (x - y)(x + y)(x - y)(x^2 + xy + y^2) = (x - y)^2(x + y)(x^2 + xy + y^2)$ 32) $a^5 - 4a^3 - 8a^2 + 32 = a^3(a^2 - 4) - 8(a^2 - 4) = (a^2 - 4)(a^3 - 8) = (a - 2)(a + 2)(a - 2)(a^2 + 2a + 4) = (a - 2)^2(a + 2)(a^2 + 2a + 4)$ 33) $27 - 27p^2 + p^3 + p^5 = 27(1 - p^2) + p^3(1 + p^2) = 27(1 - p)(1 + p) + p^3(1 + p^2) = (1+p^2)(p^3+27) - 54p^2 = (1+p^2)(p+3)(p^2-3p+9) - 54p^2$ (Группировка по $27+p^3$ и $-27p^2+p^5$): $(27+p^3) - p^2(27-p^3) = (3+p)(9-3p+p^2) - p^2(3-p)(9+3p+p^2)$ 34) $4x^4 - 0.5x = 0.5x(8x^3 - 1) = 0.5x(2x - 1)(4x^2 + 2x + 1)$ 35) $1/9y + 3y^4 = 1/9y(1 + 27y^3) = 1/9y(1 + 3y)(1 - 3y + 9y^2)$ 36) $x^6 + 2x^3y^3 + y^6 = (x^3 + y^3)^2 = ((x + y)(x^2 - xy + y^2))^2 = (x + y)^2(x^2 - xy + y^2)^2$ 37) $a^6 - 16a^3 + 64 = (a^3 - 8)^2 = ((a - 2)(a^2 + 2a + 4))^2 = (a - 2)^2(a^2 + 2a + 4)^2$ 38) $16m^4 + 2m = 2m(8m^3 + 1) = 2m(2m + 1)(4m^2 - 2m + 1)$ 39) $3k - 81k^4 = 3k(1 - 27k^3) = 3k(1 - 3k)(1 + 3k + 9k^2)$ 40) $a^6 - a^4b^2 + a^3b^3 - ab^5 = a^4(a^2 - b^2) + ab^3(a^2 - b^2) = (a^2 - b^2)(a^4 + ab^3) = (a - b)(a + b)a(a^3 + b^3) = a(a - b)(a + b)(a + b)(a^2 - ab + b^2) = a(a - b)(a + b)^2(a^2 - ab + b^2)$ 41) $(x^2 + y^2)^2 - 4x^2y^2 = (x^2 + y^2 - 2xy)(x^2 + y^2 + 2xy) = (x - y)^2(x + y)^2$ 42) $4a^2b^2 - (a^2 + b^2)^2 = (2ab - (a^2 + b^2))(2ab + (a^2 + b^2)) = -(a^2 - 2ab + b^2)(a^2 + 2ab + b^2) = -(a - b)^2(a + b)^2$ 43) $(2x - 3)^2 - 9 = (2x - 3 - 3)(2x - 3 + 3) = (2x - 6)(2x) = 4x(x - 3)$ 44) $16 - (5 - 6a)^2 = (4 - (5 - 6a))(4 + (5 - 6a)) = (4 - 5 + 6a)(4 + 5 - 6a) = (6a - 1)(9 - 6a) = 3(6a - 1)(3 - 2a)$ 45) $(4x + 7)^2 - 25x^2 = (4x + 7 - 5x)(4x + 7 + 5x) = (7 - x)(9x + 7)$ 46) $49m^2 - (3m + 4)^2 = (7m - (3m + 4))(7m + (3m + 4)) = (4m - 4)(10m + 4) = 8(m - 1)(5m + 2)$ 47) $(5x + 4)^2 - (4x - 5)^2 = (5x + 4 - 4x + 5)(5x + 4 + 4x - 5) = (x + 9)(9x - 1)$ 48) $(7 - 3p)^2 - (8p + 1)^2 = (7 - 3p - 8p - 1)(7 - 3p + 8p + 1) = (6 - 11p)(5p + 8)$ 49) $(9a + 4)^2 - (3 + 8a)^2 = (9a + 4 - 3 - 8a)(9a + 4 + 3 + 8a) = (a + 1)(17a + 7)$ 50) $4(2a + 1)^2 - 9a^2 = (2(2a + 1))^2 - (3a)^2 = (4a + 2 - 3a)(4a + 2 + 3a) = (a + 2)(7a + 2)$ 51) $36m^2 - 9(10m + 3)^2 = (6m)^2 - (3(10m + 3))^2 = (6m - 30m - 9)(6m + 30m + 9) = (-24m - 9)(36m + 9) = -9(8m + 3)(4m + 1)$ Решение уравнений: 1) $x - x^3 = 0 \Rightarrow x(1 - x^2) = 0 \Rightarrow x(1 - x)(1 + x) = 0 \Rightarrow x_1=0, x_2=1, x_3=-1$ 2) $4y^3 - y = 0 \Rightarrow y(4y^2 - 1) = 0 \Rightarrow y(2y - 1)(2y + 1) = 0 \Rightarrow y_1=0, y_2=0.5, y_3=-0.5$ 3) $9y^2 - 4y^4 = 0 \Rightarrow y^2(9 - 4y^2) = 0 \Rightarrow y^2(3 - 2y)(3 + 2y) = 0 \Rightarrow y_1=0, y_2=1.5, y_3=-1.5$ 4) $11x^2 + 88x + 176 = 0 \Rightarrow 11(x^2 + 8x + 16) = 0 \Rightarrow 11(x + 4)^2 = 0 \Rightarrow x = -4$ 5) $m^3 + 6m^2 - m - 6 = 0 \Rightarrow m^2(m + 6) - 1(m + 6) = 0 \Rightarrow (m + 6)(m^2 - 1) = 0 \Rightarrow (m + 6)(m - 1)(m + 1) = 0 \Rightarrow m_1=-6, m_2=1, m_3=-1$ 6) $x^3 - 5x^2 - 16x + 80 = 0 \Rightarrow x^2(x - 5) - 16(x - 5) = 0 \Rightarrow (x - 5)(x^2 - 16) = 0 \Rightarrow (x - 5)(x - 4)(x + 4) = 0 \Rightarrow x_1=5, x_2=4, x_3=-4$ 7) $27m - 90 = 3m^3 - 10m^2 \Rightarrow 27m - 90 - 3m^3 + 10m^2 = 0 \Rightarrow -3m(m^2 - 9) + 10(m^2 - 9) = 0 \Rightarrow (m^2 - 9)(10 - 3m) = 0 \Rightarrow (m - 3)(m + 3)(10 - 3m) = 0 \Rightarrow m_1=3, m_2=-3, m_3=10/3$ 8) $8a^3 + 12a^2 - 18a - 27 = 0 \Rightarrow 4a^2(2a + 3) - 9(2a + 3) = 0 \Rightarrow (2a + 3)(4a^2 - 9) = 0 \Rightarrow (2a + 3)(2a - 3)(2a + 3) = 0 \Rightarrow (2a + 3)^2(2a - 3) = 0 \Rightarrow a_1=-1.5, a_2=1.5$ 9) $0.125 - 0.25p - 0.5p^2 + p^3 = 0 \Rightarrow 0.125(1 - 2p) - 0.5p^2(1 - 2p) = 0 \Rightarrow (1 - 2p)(0.125 - 0.5p^2) = 0 \Rightarrow 0.125(1 - 2p)(1 - 4p^2) = 0 \Rightarrow 0.125(1 - 2p)(1 - 2p)(1 + 2p) = 0 \Rightarrow p_1=0.5, p_2=-0.5$