What's the highest power of $x$ in this nested polynomial?

What's the highest power of $x$ in this nested polynomial?

Question

The highest power of $x$ in $\left\{\left[(2x^6+7)^3+(3x^8-12)^4\right]^5 + \left[(3x^5-12x^2)^5+(x^7+6)^4\right]^6\right\}^3$ is

1. $x^{424}$,

---

1. $x^{450}$,

---

1. $x^{500}$,

---

1. $x^{504}$.

Suggestion

What is the highest power of $x$ in $(2x^6+7)^3$? Can we do the same for the other terms?

Solution

The highest power of $x$ in $(2x^6 + 7)^3$ is $x^{18}$, and in $(3x^8 - 12)^4$ is $x^{32}$, so in the first square bracket the highest power is $(x^{32})^5 = x^{160}$.

Similarly, the highest power of $x$ in $(3x^5 - 12x^2)^5$ is $x^{25}$, and in $(x^7 + 6)^4$ is $x^{28}$, so in the second square bracket the highest power is $(x^{28})^6 = x^{168}$.

So overall, the highest power of $x$ in the expansion is $(x^{168})^3 = x^{504}$, and the answer is (d).