Question

Tìm các giá trị a,b,c để:

a) \(x^4-2x^3+2x^2-2x+a=\left(x^2-2x+1\right)\left(x^2+bx+c\right)\)

b) \(x^3+3x^2-x-3=\left(x-2\right)\left(x^2+bx+c\right)+a\)

Answer 1

a: \(\left(x^2-2x+1\right)\left(x^2+bx+c\right)\)

\(=x^4+bx^3+cx^2-2x^3-2b\cdot x^2-2x\cdot c+x^2+bx+c\)

\(=x^4+x^3\left(b-2\right)+x^2\left(c-2b+1\right)+x\left(-2+b\right)+c\)

Theo đề, ta có: b-2=-2; c-2b+1=2; b-2=-2; a=c

=>b=0; c=1; a=c=1

b: (x-2)(x^2+bx+c)+a

\(=x^3+bx^2+cx-2x^2-2bx-2c+a\)

\(=x^3+x^2\left(b-2\right)+x\left(c-2b\right)-2c+a\)

Theo đề ta có: b-2=3; c-2b=-1; -2c+a=-3

=>b=5; c=-1+2b=-1+10=9; a=-3+2c=-3+2*9=15