Difference between revisions of "2017 AMC 12B Problems/Problem 23"

Problem 23

The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is 24. What is $f(0)$?

$\textbf{(A)}\quad {-2} \qquad \qquad \textbf{(B)}\quad 0 \qquad\qquad \textbf{(C)}\quad 2 \qquad\qquad \textbf{(D)}\quad \dfrac{24}5 \qquad\qquad\textbf{(E)}\quad 8$

Solution

First, we can define $f(x) = a(x-2)(x-3)(x-4) +x^2$, which contains points $A$, $B$, and $C$. Now we find that lines $AB$, $AC$, and $BC$ are defined by the equations $y = 5x - 6$, $y= 6x-8$, and $y=7x-12$ respectively. Since we want to find the $x$-coordinates of the intersections of these lines and $f(x)$, we set each of them to $f(x)$, and synthetically divide by the solutions we already know exist (eg. if we were looking at line $AB$, we would synthetically divide by the solutions $x=2$ and $x=3$, because we already know $AB$ intersects the graph at $A$ and $B$, which have $x$-coordinates of $2$ and $3$). After completing this process on all three lines, we get that the $x$-coordinates of $D$, $E$, and $F$ are $\frac{4a-1}{a}$, $\frac{3a-1}{a}$, and $\frac{2a-1}{a}$ respectively. Adding these together, we get $\frac{9a-3}{a} = 24$ which gives us $a = -\frac{1}{5}$. Substituting this back into the original equation, we get $f(x) = -\frac{1}{5}(x-2)(x-3)(x-4) + x^2$, and $f(0) = -\frac{1}{5}(-2)(-3)(-4) + 0 = \boxed{\textbf{(D)}\frac{24}{5}}$

Solution 2

No need to find the equations for the lines, really. First of all, $f(x) = a(x-2)(x-3)(x-4) +x^2$. Let's say the line $AB$ is $y=bx+c$, and $x_1$ is the $x$ coordinate of the third intersection, then $2$, $3$, $x_1$ are the three roots of $f(x) - bx-c$. Apparently the value of $b$ and $c$ have no effect on the sum of the 3 roots, because the coefficient of the $x^2$ term is always $9a-1$. So we have, $$\frac{9a-1}{a} = 2+3 + x_1=3+4+x_2 = 2+4+x_3$$ Add them up we have $$3\frac{9a-1}{a} = 18 + x_1+x_2+x_3 = 18 +24$$ Solve it, we get $a = -\frac{1}{5}$. $\boxed{\textbf{(D)}\frac{24}{5}}$.

- Mathdummy