Problem of the Day


If f(x)=ax^5+bx^4+cx^3+dx^2+ex+f, what is the value of a+b+c+d+e+f?

(1) The graphs of y=5, y=3x+2, and y=f(x) all intersect at the same point on the xy-coordinate plane.

(2) (a+b)(c+d)(e+f)=0

Reveal Answer

Answer

A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

See the Solution

Solution

[latexpage]

The question is asking for the sum of the coefficients of $f(x)$. Note that this is the same as $f(1)$.

Statement (1): SUFFICIENT. We know that all three graphs intersect at the same point. Two distinct lines can only intersect at one point, so that must be the referenced intersection point. $y=5$ and $y=3x+2$ intersect at the point $(1,5)$. If this point is also on the graph of $y=f(x)$, then $f(1)=5$.

Statement (2): INSUFFICIENT. This statement tells us that at least one of the following is true: $a+b=0, c+d=0$ or $e+f=0$. However, we do not know which ones are true. If they were all true, we could determine that the sum of the coefficients is zero. However, as it stands, there is not enough information to conclude this.


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