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

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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

$f(x)$ 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

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

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

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

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 $f(1) = 1 + 1 + (-8009) + 100 + 900 = \boxed{-7007}$ or $\boxed{\text{C}}$ .

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

@@ Line 56: / Line 56: @@
 We know that <math>f(1)</math> is the sum of its coefficients, hence <math>1+1+b+100+c</math>. We substitute the values we obtained for <math>b</math> and <math>c</math> into this expression to get <math>f(1) = 1 + 1 + (-8009) + 100 + 900 = \boxed{-7007}</math> or <math>\boxed{\text{C}}</math>.
+==Solution 3==
+Let <math>r_1,r_2,</math> and <math>r_3</math> be the roots of <math>g(x)</math>. Let <math>r_4</math> be the additional root of <math>f(x)</math>. Then from Vieta's formulas on the quadratic term of <math>g(x)</math> and the cubic term of <math>f(x)</math>, we obtain the following:
+<cmath>\begin{align*}
+r_1+r_2+r_3&=-a \\
+r_1+r_2+r_3+r_4&=-1
+\end{align*}</cmath>
+Thus <math>r_4=a-1</math>.
+Now applying Vieta's formulas on the constant term of <math>g(x)</math>, the linear term of <math>g(x)</math>, and the linear term of <math>f(x)</math>, we obtain:
+<cmath>\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*}</cmath>
+Substituting for <math>r_1r_2r_3</math> in the bottom equation and factoring the remainder of the expression, we obtain:
+<cmath>-10+(r_1r_2+r_2r_3+r_3r_1)r_4=-10+r_4=-100</cmath>
+It follows that <math>r_4=-90</math>. But <math>r_4=a-1</math> so <math>a=-89</math>
+Now we can factor <math>f(x)</math> in terms of <math>g(x)</math> as
+<cmath>f(x)=(x-r_4)g(x)=(x+90)g(x)</cmath>
+Then <math>f(1)=91g(1)</math> and
+<cmath>g(1)=1^3-89\cdot 1^2+1+10=-77</cmath>
+Hence <math>f(1)=91\cdot(-77)=\boxed{\textbf{(C)}\,-7007}</math>.
 ==See Also==
 {{AMC10 box|year=2017|ab=A|num-b=8|num-a=10}}
 {{MAA Notice}}

2017 AMC 10A (Problems • Answer Key • Resources)
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