最後三題因式分解有水準
9.已知x+3是多項式x^3+x^2+mx+3的因式
(a)求m的值
(b)將該多項式因式分解
10.已知x-4是多項式x^3+Cx^2-x-C的因式
(a)求C的值
(b)將該多項式因式分解
21.已知f(x)=8x^3+(4k+4)^2-4x+k，其中k≠0。若f(x)可被2x+k整除
[a]求k的值
[b]將f(x)因式分解
22.已知x+1和x-2都是x^3-ax^2+x-b的因式
[a]求a和b的值
[b]求該多項式其餘的因式
26. 8x^4-28x^3+16x^2-2x
27. x^4-5x^3-2x^2+9x+5
28. x^4+2x^3-7x^2-8x+12
9.
(a)令x=-3代入, -27+9-3m+3=0, m=-5
(b)得原式為x^3+x^2-5x+3
=(x+3)(x^2-2x+1)=(x+3)(x-1)^2
10.
(a)令x=4,代入得4^3+4^2C-4-C=0, C=-4
(b)得原式為: x^3+12x^2-x-12
(x-4)(x^2-1)= (x-4)(x+1)(x-1)
21. [a] 由長除法得:
[8x^3+(4k+4)x^2-4x+k]/(2x+k)=(4x^2+2x+1)….0
得 k=-3,
[b]f(x)= (4x^2+2x+1)(2x-3)
22.
[a](x+1)(x-2)=x^2-x-2
由長除法得: (x^3-ax^2+x-b)/( x^2-x-2)=(x-3)…….0
b=-6, a=4
[b]其餘的因式: (x-3)
長除法過程
26.
8x^4-28x^3+16x^2-2x
=2x(4x^3-14x^2+8x-1)
=(2x-1)(2x^2-6x+1)
8x^4-28x^3+16x^2-2x
=2x(4x^3-14x^2+8x-1)
=(2x-1)(2x^2-6x+1)
27.
x^4-5x^3-2x^2+9x+5
=x^3(x-5)-(2x^2-9x-5)
=x^3(x-5)-(2x+1)(x-5)
=(x-5)(x^3-2x-1)
=(x-5)(x+1)(x^2-x-1)
x^4-5x^3-2x^2+9x+5
=x^3(x-5)-(2x^2-9x-5)
=x^3(x-5)-(2x+1)(x-5)
=(x-5)(x^3-2x-1)
=(x-5)(x+1)(x^2-x-1)
28.
x^4+2x^3-7x^2-8x+12
=x^4+2x^3+ax^2+bx^2-8x+12
(1:2:a=b:-8:12, a=-3, b=-4)
=x^4+2x^3-3x^2-4x^2-8x+12
=x^2(x^2+2x-3)-4(x^2+2x-3)
=(x^2-4)(x^2+2x-3)
=(x+2)(x-2)(x^2+2x-3)
x^4+2x^3-7x^2-8x+12
=x^4+2x^3+ax^2+bx^2-8x+12
(1:2:a=b:-8:12, a=-3, b=-4)
=x^4+2x^3-3x^2-4x^2-8x+12
=x^2(x^2+2x-3)-4(x^2+2x-3)
=(x^2-4)(x^2+2x-3)
=(x+2)(x-2)(x^2+2x-3)