Question

O coeficiente de x⁴ no polinômio P(x)=(x+2)^6 é?

Answer 1

Da definição de binômio de Newton, temos que:

$\\ (a + b)^n = \sum_{j = 0}^{n} \binom{n}{j} \cdot a^{n - j} \cdot b^j \\\\\\ P(x) = \sum_{j = 0}^{6}\binom{6}{j} a^{6 - j} \cdot b^j$

 Com efeito, o coeficiente de $x^4$ será dado por $\binom{6}{j} \cdot (x)^{6 - j} \cdot (2)^j$. Isto é,

$\\ 6 - j = 4 \\ - j = 4 - 6 \\ \boxed{j = 2}$

 Por fim,

$\\ \binom{6}{j} \cdot (x)^{6 - j} \cdot (2)^j = \\\\\\ \binom{6}{2} \cdot (x)^{6 - 2} \cdot (2)^2 = \\\\\\ C_{6, 2} \cdot x^4 \cdot 4 = \\\\\\ \frac{6!}{(6 - 2)!2!} \cdot 4x^4 = \\\\\\ \frac{6 \cdot 5 \cdot 4!}{4!2 \cdot 1} \cdot 4x^4 = \\\\\\ \boxed{\boxed{60}x^4}$