Possible Duplicate:
What is shortcut to this contest algebra problem about polynomial?
The polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ has the property that $P(k)=11k$ for $k=1,2,3,4$. Compute $c$.
Possible Duplicate:
What is shortcut to this contest algebra problem about polynomial?
The polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ has the property that $P(k)=11k$ for $k=1,2,3,4$. Compute $c$.
$P(x)-11x=(x-1)(x-2)(x-3)(x-4)$ since it is monic, has degree four, and is zero at $1,2,3,4$. the coefficient of $x$ is $-(1\cdot2\cdot3+1\cdot2\cdot4+1\cdot3\cdot4+2\cdot3\cdot4)=-(6+8+12+24)=-50$. hence the coefficient of $x$ in $P(x)$ is $c=-39$. likewise, $a=-(4+3+2+1)=-10$, $b=1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4=2+3+4+6+8+12=35$ and $d=1\cdot2\cdot3\cdot4=24$ so that $P(x)=x^4-10x^3+35x^2-39x+24$