There's fifth degree polynomial, it's first coefficient equals $-7$.
$$-7x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$$
Also:
$$W(1)=-2$$
$$W(2)=-4$$
$$W(3)=-6$$
$$W(4)=-8$$
$$W(5)=-10$$
Find the value of constant term.
It could be solved by system of equations.
But I think that there's an easier way to do it.
I've tried to sum some of the given values, and erase other coefficients, but I`m not sure it leads somewhere.
Could someone help me solve this and help me to understand it?
Best Answer
The obvious linear function fitting the five given points is $-2x$. We split that out from the polynomial: $$W(x)=[-7x^5+a_4x^4+a_3x^3+a_2x^2+(a_1+2)x+a_0]-2x$$ It is clear that the square-bracketed expression must be 0 at $x=1,2,3,4,5$, so can be written as $$-7(x-1)(x-2)(x-3)(x-4)(x-5)$$ whose constant term, and thus $a_0$, is $(-7)(-1)(-2)(-3)(-4)(-5)=840$.