Question

[addfx]

xác định hệ số a b c sao cho đa thức f(x)= 2x^4+ax^2 +bx+c chia hết cho x-2 khi chia f(x) cho x^2-4x+3 thì được phần dư là -x+2

 

Answer 1

[Nguyễn Lê Phước Thịnh]

\(f\left(x\right)=2x^4+ax^2+bx+c\)

\(=2x^4-4x^3+4x^3-8x^2+\left(a+8\right)x^2-x\left(2a+16\right)+\left(2a+16+b\right)x-2\left(2a+16+b\right)+4a+32+2b+c\)

\(=\left(x-2\right)\left(2x^3+4x^2+x\left(a+8\right)+2a+16+b\right)+4a+2b+32+c\)

=>\(\dfrac{f\left(x\right)}{x-2}=2x^3+4x^2+x\left(a+8\right)+2a+16+b+\dfrac{4a+2b+32+c}{x-2}\)

f(x) chia hết cho x-2 nên \(4a+2b+32+c=0\)(1)

\(f\left(x\right)=2x^4+ax^2+bx+c\)

\(=2x^4-4x^3+6x^2+4x^3-16x^2+12x+\left(a+10\right)x^2-4x\left(a+10\right)+3a+30+x\left(4a+28+b\right)+c-3a-30\)

\(=\left(x^2-4x+3\right)\left(2x^2+4x+a+10\right)\)+x(4a+28+b)+c-3a-30

f(x) chia cho x²-4x+3 dư -x+2 nên ta có: 

\(\left\{{}\begin{matrix}4a+28+b=-1\\c-3a-30=2\end{matrix}\right.\)(2)

Từ (1),(2) ta có hệ phương trình:

\(\left\{{}\begin{matrix}4a+2b+32+c=0\\4a+b+28=-1\\c-3a=32\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}4a+2b+c=-32\\4a+b=-29\\-3a+c=32\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b+c=-3\\-3a+c=32\\4a+b=-29\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}b+3a=-35\\4a+b=-29\\b+c=-3\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}-a=-6\\4a+b=-29\\b+c=-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=6\\b=-29-4a=-29-4\cdot6=-53\\c=-3-b=-3-\left(-53\right)=50\end{matrix}\right.\)