Difference between revisions of "2021 AMC 10B Problems/Problem 15"

Problem

The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3?$

$\textbf{(A)} ~-1 \qquad\textbf{(B)} ~0 \qquad\textbf{(C)} ~1 \qquad\textbf{(D)} ~2 \qquad\textbf{(E)} ~\sqrt{5}$

Solution 1

We square $x+\frac{1}{x}=\sqrt5$ to get $x^2+2+\frac{1}{x^2}=5$. We subtract 2 on both sides for $x^2+\frac{1}{x^2}=3$ and square again, and see that $x^4+2+\frac{1}{x^4}=9$ so $x^4+\frac{1}{x^4}=7$. We can divide our original expression of $x^{11}-7x^7+x^3$ by $x^7$ to get that it is equal to $x^7(x^4-7+\frac{1}{x^4})$. Therefore because $x^4+\frac{1}{x^4}$ is 7, it is equal to $x^7(0)=\boxed{(B) 0}$.

Solution 2

Multiplying both sides by $x$ and using the quadratic formula, we get $\frac{\sqrt{5} \pm 1}{2}$. We can assume that it is $\frac{\sqrt{5}+1}{2}$, and notice that this is also a solution the equation $x^2-x-1=0$, i.e. we have $x^2=x+1$. Repeatedly using this on the given (you can also just note Fibonacci numbers), \begin{align*} (x^{11})-7x^7+x^3 &= (x^{10}+x^9)-7x^7+x^3 \\ &=(2x^9+x^8)-7x^7+x^3 \\ &=(3x^8+2x^7)-7x^7+x^3 \\ &=(3x^8-5x^7)+x^3 \\ &=(-2x^7+3x^6)+x^3 \\ &=(x^6-2x^5)+x^3 \\ &=(-x^5+x^4+x^3) \\ &=-x^3(x^2-x-1) = \boxed{(\textbf{B}) 0} \end{align*}

~Lcz

Solution 3

We can immediately note that the exponents of $x^{11}-7x^7+x^3$ are an arithmetic sequence, so they are symmetric around the middle term. So, $x^{11}-7x^7+x^3 = x^7(x^4-7+\frac{1}{x^4})$. We can see that since $x+\frac{1}{x} = \sqrt{5}$, $x^2+2+\frac{1}{x^2} = 5$ and therefore $x^2+\frac{1}{x^2} = 3$. Continuing from here, we get $x^4+2+\frac{1}{x^4} = 9$, so $x^4-7+\frac{1}{x^4} = 0$. We don't even need to find what $x^3$ is! This is since $x^3\cdot0$ is evidently $\boxed{(B) 0}$, which is our answer.

~sosiaops

Solution 4

We begin by multiplying $x+\frac{1}{x} = \sqrt{5}$ by $x$, resulting in $x^2+1 = \sqrt{5}x$. Now we see this equation: $x^{11}-7x^{7}+x^3$. The terms all have $x^3$ in common, so we can factor that out, and what we're looking for becomes $x^3(x^8-7x^4+1)$. Looking back to our original equation, we have $x^2+1 = \sqrt{5}x$, which is equal to $x^2 = \sqrt{5}x-1$. Using this, we can evaluate $x^4$ to be $5x^2-2\sqrt{5}x+1$, and we see that there is another $x^2$, so we put substitute it in again, resulting in $3\sqrt{5}x-4$. Using the same way, we find that $x^8$ is $21\sqrt{5}x-29$. We put this into $x^3(x^8-7x^4+1)$, resulting in $x^3(0)$, so the answer is $\boxed{(B)~0}$.

~purplepenguin2

Solution 5

The equation we are given is $x+\tfrac{1}{x}=\sqrt{5}...$ Yuck. Fractions and radicals! We multiply both sides by $x,$ square, and re-arrange to get $$x^2+1=\sqrt{5}x \implies x^4+2x^2+1=5x^2 \implies x^4-3x^2+1=0.$$ Now, let us consider the expression we wish to acquire. Factoring out $x^3,$ we have $$x^3\left(x^8-7x^4+1\right) = x^3\left(x^8+2x^4+1-9x^4\right).$$ Then, we notice that $x^8+2x^4+1=\left(x^4+1\right)^2.$ Furthermore, $$x^4+1=3x^2 \implies \left(x^4+1\right)^2=x^8+2x^4+1=9x^4.$$ Thus, our answer is $$x^3\left(9x^4-9x^4\right) = x^3 \cdot 0 = \boxed{\textbf{(B)}} ~ 0.$$

~ pi_is_3.14

~ Interstigation

Video Solution by TheBeautyofMath

Not the most efficient method, but gets the job done.

~IceMatrix