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?
$\endgroup$ 21 Answer
$\begingroup$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 $(-7)(-1)(-2)(-3)(-4)(-5)=840$.
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