Difference between revisions of "2017 AMC 10A Problems/Problem 24"

m (Solution 4 (Slight guessing))
(Better solution ending to solution 1.1.)
Line 22: Line 22:
 
-10r&=c.\\
 
-10r&=c.\\
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
 +
Solution 1.1 picks up here.
  
 
Let's solve for <math>a,b,c,</math> and <math>r</math>. Since <math>10-r=100</math>, <math>r=-90</math>, so <math>c=(-10)(-90)=900</math>. Since <math>a-r=1</math>, <math>a=-89</math>, and <math>b=1-ar=-8009</math>. Thus, we know that
 
Let's solve for <math>a,b,c,</math> and <math>r</math>. Since <math>10-r=100</math>, <math>r=-90</math>, so <math>c=(-10)(-90)=900</math>. Since <math>a-r=1</math>, <math>a=-89</math>, and <math>b=1-ar=-8009</math>. Thus, we know that
Line 35: Line 37:
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  
 +
==Solution 1.1==
 +
A faster ending to Solution 1 is as follows.
 +
We shall solve for only <math>a</math> and <math>r</math>. Since <math>10-r=100</math>, <math>r=-90</math>, and since <math>a-r=1</math>, <math>a=-89</math>. Then,
 +
<cmath>\begin{align*}
 +
f(1)&=(1+r)(x^3+ax^2+x+10)\\
 +
&=(91)(-77)\\
 +
&=\boxed{\bold{(C)}\, -7007}.\\
 +
\end{align*}</cmath>
 +
Thanks for reading, Rowechen Zhong.
 
==Solution 2==
 
==Solution 2==
 
We notice that the constant term of <math>f(x)=c</math> and the constant term in <math>g(x)=10</math>. Because <math>f(x)</math> can be factored as <math>g(x) \cdot (x- r)</math> (where <math>r</math> is the unshared root of <math>f(x)</math>, we see that using the constant term, <math>-10 \cdot r = c</math> and therefore <math>r = -\frac{c}{10}</math>.
 
We notice that the constant term of <math>f(x)=c</math> and the constant term in <math>g(x)=10</math>. Because <math>f(x)</math> can be factored as <math>g(x) \cdot (x- r)</math> (where <math>r</math> is the unshared root of <math>f(x)</math>, we see that using the constant term, <math>-10 \cdot r = c</math> and therefore <math>r = -\frac{c}{10}</math>.

Revision as of 16:28, 23 May 2017

Problem

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\]What is $f(1)$?

$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$

Solution 1

$f(x)$ must have four roots, three of which are roots of $g(x)$. Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficient of $f(x)$ and $g(x)$ are the same, we know that

\[f(x)=g(x)(x-r)\]

where $r\in\mathbb{C}$ is the fourth root of $f(x)$. Substituting $g(x)$ and expanding, we find that

\begin{align*}f(x)&=(x^3+ax^2+x+10)(x-r)\\ &=x^4+(a-r)x^3+(1-ar)x^2+(10-r)x-10r.\end{align*}

Comparing coefficients with $f(x)$, we see that

\begin{align*} a-r&=1\\ 1-ar&=b\\ 10-r&=100\\ -10r&=c.\\ \end{align*}

Solution 1.1 picks up here.

Let's solve for $a,b,c,$ and $r$. Since $10-r=100$, $r=-90$, so $c=(-10)(-90)=900$. Since $a-r=1$, $a=-89$, and $b=1-ar=-8009$. Thus, we know that

\[f(x)=x^4+x^3-8009x^2+100x+900.\]

Taking $f(1)$, we find that

\begin{align*} f(1)&=1^4+1^3-8009(1)^2+100(1)+900\\ &=1+1-8009+100+900\\ &=\boxed{\bold{(C)}\, -7007}.\\ \end{align*}

Solution 1.1

A faster ending to Solution 1 is as follows. We shall solve for only $a$ and $r$. Since $10-r=100$, $r=-90$, and since $a-r=1$, $a=-89$. Then, \begin{align*} f(1)&=(1+r)(x^3+ax^2+x+10)\\ &=(91)(-77)\\ &=\boxed{\bold{(C)}\, -7007}.\\ \end{align*} Thanks for reading, Rowechen Zhong.

Solution 2

We notice that the constant term of $f(x)=c$ and the constant term in $g(x)=10$. Because $f(x)$ can be factored as $g(x) \cdot (x- r)$ (where $r$ is the unshared root of $f(x)$, we see that using the constant term, $-10 \cdot r = c$ and therefore $r = -\frac{c}{10}$. Now we once again write $f(x)$ out in factored form:

\[f(x) = g(x)\cdot (x-r) = (x^3+ax^2+x+10)(x+\frac{c}{10})\].

We can expand the expression on the right-hand side to get:

\[f(x) = x^4+(a+\frac{c}{10})x^3+(1+\frac{ac}{10})x^2+(10+\frac{c}{10})x+c\]

Now we have $f(x) = x^4+(a+\frac{c}{10})x^3+(1+\frac{ac}{10})x^2+(10+\frac{c}{10})x+c=x^4+x^3+bx^2+100x+c$.

Simply looking at the coefficients for each corresponding term (knowing that they must be equal), we have the equations: \[10+\frac{c}{10}=100 \Rightarrow c=900\] \[a+\frac{c}{10} = 1, c=900 \Rightarrow a + 90 =1 \Rightarrow a= -89\]

and finally,

\[1+\frac{ac}{10} = b = 1+\frac{-89 \cdot 900}{10} = b = -8009\].

We know that $f(1)$ is the sum of its coefficients, hence $1+1+b+100+c$. We substitute the values we obtained for $b$ and $c$ into this expression to get $f(1) = 1 + 1 + (-8009) + 100 + 900 = \boxed{\textbf{(C)}\,-7007}$.

Solution 3

Let $r_1,r_2,$ and $r_3$ be the roots of $g(x)$. Let $r_4$ be the additional root of $f(x)$. Then from Vieta's formulas on the quadratic term of $g(x)$ and the cubic term of $f(x)$, we obtain the following:

\begin{align*} r_1+r_2+r_3&=-a \\  r_1+r_2+r_3+r_4&=-1 \end{align*}

Thus $r_4=a-1$.

Now applying Vieta's formulas on the constant term of $g(x)$, the linear term of $g(x)$, and the linear term of $f(x)$, we obtain:

\begin{align*} r_1r_2r_3  & = -10\\ r_1r_2+r_2r_3+r_3r_1 &= 1\\  r_1r_2r_3+r_2r_3r_4+r_3r_4r_1+r_4r_1r_2  & = -100\\ \end{align*}

Substituting for $r_1r_2r_3$ in the bottom equation and factoring the remainder of the expression, we obtain:

\[-10+(r_1r_2+r_2r_3+r_3r_1)r_4=-10+r_4=-100\]

It follows that $r_4=-90$. But $r_4=a-1$ so $a=-89$

Now we can factor $f(x)$ in terms of $g(x)$ as

\[f(x)=(x-r_4)g(x)=(x+90)g(x)\]

Then $f(1)=91g(1)$ and

\[g(1)=1^3-89\cdot 1^2+1+10=-77\]

Hence $f(1)=91\cdot(-77)=\boxed{\textbf{(C)}\,-7007}$.

Solution 4 (Slight guessing)

Let the roots of $g(x)$ be $r_1$, $r_2$, and $r_3$. Let the roots of $f(x)$ be $r_1$, $r_2$, $r_3$, and $r_4$. From Vieta's, we have: \begin{align*} r_1+r_2+r_3=-a \\ r_1+r_2+r_3+r_4=-1 \\ r_4=a-1 \end{align*} The fourth root is $a-1$. Since $r_1$, $r_2$, and $r_3$ are common roots, we have: \begin{align*} f(x)=g(x)(x-(a-1)) \\ f(1)=g(1)(1-(a-1)) \\ f(1)=(a+12)(2-a) \\ f(1)=-(a+12)(a-2) \\ \end{align*} Let $a-2=k$: \begin{align*} f(1)=-k(k+14) \end{align*} Note that $-7007=-1001\cdot(7)=-(7\cdot(11)\cdot(13))\cdot(7)=-91\cdot(77)$ This gives us a pretty good guess of $\boxed{\textbf{(C)}\, -7007}$.

Solution 5

First off, let's get rid of the $x^4$ term by finding $h(x)=f(x)-xg(x)$. This polynomial consists of the difference of two polynomials with $3$ common factors, so it must also have these factors. The polynomial is $h(x)=(1-a)x^3 + (b-1)x^2 + 90x + c$, and must be equal to $(1-a)g(x)$. Equating the coefficients, we get $4$ equations. We will tackle the situation one equation at a time, starting the $x$ terms. \[90=1-a\] The solution is obviously $a=-89$. We can now find b and c with ease. \[b-1=a(1-a)=-89*90=-8010\] and $b=-8009$. \[c=10(1-a)=10*90=900\] Finally, $f(1)=102+b+c=102+900-8009=\boxed{\textbf{(C)}\, -7007}$.

See Also

2017 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2017 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS