Difference between revisions of "2007 AIME I Problems/Problem 8"

Revision as of 12:09, 4 July 2020

Problem

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$ ?

Solution

Solution 1

We can see that $Q_1$ and $Q_2$ must have a root in common for them to both be factors of the same cubic.

Let this root be $a$ .

We then know that $a$ is a root of $Q_{2}(x)-2Q_{1}(x) = 2x^{2}+2kx-43x+k-2x^{2}-2kx+58x+2k = 15x+3k = 0$ , so $x = \frac{-k}{5}$ .

We then know that $\frac{-k}{5}$ is a root of $Q_{1}$ so we get: $\frac{k^{2}}{25}+(k-29)\left(\frac{-k}{5}\right)-k = 0 = k^{2}-5(k-29)(k)-25k = k^{2}-5k^{2}+145k-25k$ or $k^{2}=30k$ , so $k=30$ is the highest.

We can trivially check into the original equations to find that $k=30$ produces a root in common, so the answer is $030$ .

Solution 2

Again, let the common root be $a$ ; let the other two roots be $m$ and $n$ . We can write that $(x - a)(x - m) = x^2 + (k - 29)x - k$ and that $2(x - a)(x - n) = 2\left(x^2 + \left(k - \frac{43}{2}\right)x + \frac{k}{2}\right)$ .

Therefore, we can write four equations (and we have four variables), $a + m = 29 - k$ , $a + n = \frac{43}{2} - k$ , $am = -k$ , and $an = \frac{k}{2}$ .

The first two equations show that $m - n = 29 - \frac{43}{2} = \frac{15}{2}$ . The last two equations show that $\frac{m}{n} = -2$ . Solving these show that $m = 5$ and that $n = -\frac{5}{2}$ . Substituting back into the equations, we eventually find that $k = 30$ .

Solution 3

Since $Q_1(x)$ and $Q_2(x)$ are both factors of $P(x)$ , which is cubic, we know the other factors associated with each of $Q_1(x)$ and $Q_2(x)$ must be linear. Let $Q_1(x)R(x) = Q_2(x)S(x) = P(x)$ , where $R(x) = ax + b$ and $S(x) = cx + d$ . Then we have that $((x^2 + (k-29)x - k))(ax + b) = ((2x^2+ (2k-43)x + k))(cx + d)$ . Equating coefficients, we get the following system of equations:

$\begin{align} a = 2c \\ b = -d \\ 2c(k - 29) - d = c(2k - 43) + 2d \\ -d(k - 29) - 2ck = d(2k - 43) + ck \end{align}$

Using equations $(1)$ and $(2)$ to make substitutions into equation $(3)$ , we see that the $k$ 's drop out and we're left with $d = -5c$ . Substituting this expression for $d$ into equation $(4)$ and solving, we see that $k$ must be $\boxed {30}$ .

~ anellipticcurveoverq

@@ Line 29: / Line 29: @@
 The first two equations show that <math>m - n = 29 - \frac{43}{2} = \frac{15}{2}</math>. The last two equations show that <math>\frac{m}{n} = -2</math>. Solving these show that <math>m = 5</math> and that <math>n = -\frac{5}{2}</math>. Substituting back into the equations, we eventually find that <math>k = 30</math>.
+=== Solution 3 ===
+Since <math>Q_1(x)</math> and <math>Q_2(x)</math> are both factors of <math>P(x)</math>, which is cubic, we know the other factors associated with each of <math>Q_1(x)</math> and <math>Q_2(x)</math> must be linear. Let <math>Q_1(x)R(x) = Q_2(x)S(x) = P(x)</math>, where <math>R(x) = ax + b</math> and <math>S(x) = cx + d</math>. Then we have that <math>((x^2 + (k-29)x - k))(ax + b) = ((2x^2+ (2k-43)x + k))(cx + d)</math>. Equating coefficients, we get the following system of equations:
+<cmath>\begin{align}
+a = 2c \\
+b = -d \\
+c(k - 29) - d = c(2k - 43) + 2d \\
+-d(k - 29) - 2ck = d(2k - 43) + ck
+\end{align}</cmath>
+Using equations <math>(1)</math> and <math>(2)</math> to make substitutions into equation <math>(3)</math>, we see that the <math>k</math>'s drop out and we're left with <math>d = -5c</math>. Substituting this expression for <math>d</math> into equation <math>(4)</math> and solving, we see that <math>k</math> must be <math>\boxed {30}</math>.
+~ anellipticcurveoverq
 == See also ==

2007 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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