maths F.3

1) X 4次﹣10X2次+9
x^4 ﹣10 x^2 + 9
= (x^2 - 1)(x^2 - 9)
= (x^2 - 1^2)(x^2 - 3^2)
= (x - 1)(x + 1) (x - 3)(x + 3)

2) (a+b)3次+(a-b)3次
(a+b)^3 + (a-b)^3
= [(a+b) + (a-b)] [(a+b)^2 - (a+b)(a-b) + (a-b)^2]
= (2a) [(a+b)^2 - (a+b)(a-b) + (a-b)^2]
= (2a) [2a^2 + 2b^2 - (a+b)(a-b) ]
= (2a) [2a^2 + 2b^2 - (a^2-b^2) ]
= (2a) (a^2 + 3b^2)

3) 若x2次+cx+16能分解為因式，寫出c的4個可能性
x^2 + cx + 16
delta = c^2 - (4)(16) = c^2 - 64
能分解為因式, delta 要某數既平方
c^2 - 64 = 0 or c^2 - 64 = 36
c = 8, -8, 10, -10

1) x^4-10x^2+9
=(x^2-1)(x^2-9)
=(x+1)(x-1)(x+3)(x-3)

2) (a+b)^3+(a-b)^3
=a^3+3a^2 b+3ab^2+b^3+a^3-3a^2b+3ab^2-b^3
=2a^3+6ab^2
=2a(a^2+3b^2)

3)x^2+cx+16
16=2x2x2x2
(x+1)(x+16) c=17
(x+2)(x+8) c=10
(x+4)(x+4) c=8
(x-1)(x-16) c= -17
(x-2)(x-8) c= - 10
(x-4)(x-4) c= - 8

仲有d 小數...

1. x^4- 10x^2 + 9
= (x^2 -1)(x^2 -9)
=(x-1)(x+1)(x+3)(x-3)

2. (a+b)^3 + (a-b)^3
=(a+b+a-b) [(a+b)^2 - (a+b)(a-b)+ (a-b)^2]
=2a(a^2 + 3b^2)

3. x^2+ cx+ 16 = (x+ a)(x+ b)
= x^2+ (a+ b)x + ab
比較系數
ab= 16, a+b = c

所以 當a= 1 , b= 16 , then c= 17
a= 2 , b= 8 , then c= 10
a= 4 , b= 4 , then c= 8
a= -1 , b= -16 , then c= -17

因此 c= 17, 10, 8 或 -17....