Factorization is an art (4)

Algebra Level 5

$\large f(x)= x^3+21x^2+147x+336$ Let $f: \mathbb {R \to R}$ be defined as above.

If the real root of the equation $\underbrace{f(f(f(\dots f(f(x))\dots)))}_{2016\ \text{times}} =0$ is of the form $\large -a+a^{1/ {3^b}}$ , where $a$ and $b$ are integer , find $a+b$ .

The answer is 2023.

1 solution

Tommy Li
Jul 4, 2016

$f(x)= x^3+21x^2+147x+336$

$f(x)=(x+7)^{3}-7$

$f(f(x))=((x+7)^{3}-7+7)^{3}-7$

$f(f(x))=(x+7)^{3^{2}}-7$

$\large \Rightarrow \underbrace{f(f(f(\dots f(f(x))\dots)))}_{2016\ \text{times}} = (x+7)^{3^{2016}}-7$

$\large \underbrace{f(f(f(\dots f(f(x))\dots)))}_{2016\ \text{times}} =0$

$\large (x+7)^{3^{2016}}-7 =0$

$\large \Rightarrow x = -7+7^{\dfrac{1}{3^{2016}}}$

$\large \Rightarrow a+b =2016+7=2023$

$\large \Rightarrow \underbrace{f(f(f(\dots f(f(x))\dots)))}_{n \ \text{times}} = (x+7)^{3^{n}}-7$

You need to prove that the equation above holds true for ALL positive integers first. Hint : induction.

Pi Han Goh - 4 years, 11 months ago

Factorization is an art (4)

The answer is 2023.

1 solution

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