MATH FACTORIZE

(1) c² (c+2)-c-2
(2) a³-4a-45a
(3) (a²-2ab+b²)-25
(4) (k+1)²-(k+1)-56
(5) (2x-3)²+10x-15
(6) 3(x+7)²+(x+7)(x-6)-14(x-6)²

(1)
c²(c+2) – c – 2
= c³ + 2c² - c – 2
= c(c² + 2c – 1) – 2

(2)
a³ – 4a – 45a
= a(a² - 4 – 45)
= a(a² - 49)
= a(a+7)(a-7)

(3)
a² - 2ab + b² - 25
= (a-b) ² - 25
= (a-b-5)(a-b+5)

(4)
(k+1) ² - (k+1) – 56
= (k+1+7)(k+1-8)
= (k+8)(k-7)

(5)
(2x -3) ² + 10x – 15
= 4x² - 12x + 9 + 10x – 15
= 4x² - 2x – 6
= 2(2x² - x – 3)
= 2(x+1)(2x – 3)

(6)
3(x+7)²+(x+7)(x-6)-14(x-6)²
Let y = (x+7) and z = (x-6)
3y² + yz – 14z²
= (y -2z)(3y + 7z)
= (x + 7 – 2(x-6) )*(3x + 21 + 7(x-6) )
= (x + 7 – 2x + 12) (3x + 21 + 7x – 42)
= (-x + 19)(10x – 32)
= (10x – 32)(19 –x)

2006-10-30 10:23:34 補充：
(1)c²(c+2) – c – 2= c³ + 2c² - c – 2= c(c² + 2c – 1) – 2做錯=.=改正：c²(c+2) – (c+2)= (c+2)(c² - 1)= (c+2)(c+1)(c-1)

(1) c^2(c+2)-c-2
(2) a^3-4a-45a
(3) (a^2-2ab+b^2)-25
(4) (k+1)^2-(k+1)-56
(5) (2x-3)^2+10x-15
(6) 3(x+7)^2+(x+7)(x-6)-14(x-6)^2


1)
c^2 (c + 2) - c - 2
=c^2 (c + 2) - (c + 2)
=(c + 2) [(c^2) - (1)] ...........................{ 抽common factor }
=(c + 2) (c^2 - 1^2) ...........................{ 1^2 = 1}
=(c + 2) (c - 1) (c + 1) ...........................{ (a^2 - b^2) = (a - b) (a + b) }

2)
a^3 - 4a - 45a
=a^3 - 49a
=a (a^2 - 49)
=a (a^2 - 7^2)
=a (a - 7) (a + 7) ...........................{ (a^2 - b^2) = (a - b) (a + b) }

3)
(a^2 - 2ab + b^2) - 25
=(a - b)^2 - 25 ...........................{ (a - b)^2 = (a^2 - 2ab + b^2) }
=(a - b)^2 - 5^2
=[(a - b) - 5] [(a - b) + 5] ...........................{ (a^2 - b^2) = (a - b) (a + b) }
=(a - b - 5) (a - b + 5)

4)
(k + 1)^2 - (k + 1) - 56
=[(k + 1) + 7] [(k + 1) - 8]
=(k+ 8)(k - 7)

5)
(2x - 3)^2 + 10x - 15
=(2x - 3) (2x - 5) + 5(2x - 3)
=(2x - 3) [(2x - 3) + 5] ...........................{ 抽common factor }
=(2x - 3) (2x + 2)
=2(2x - 3) (x + 1)

6)
3(x + 7)^2 + (x + 7) (x - 6) - 14(x - 6)^2

Let a = (x + 7) and b=(x - 6)

=3a^2 + ab -14b^2
=(a - 2b) (3a + 7b)
=[(x + 7) - 2(x - 6)] [3(x + 7) + 7(x - 6)]
=(x + 7 - 2x + 12) (3x + 21 + 7x - 42)
=(-x + 19) (10x - 21)
=-(x - 19) (10x - 21)