Find P Coefficient 76769E

1. **State the problem:** We need to find the value of $p$ such that in the expansion of $$px^2(5 + px)^8$$ the coefficient of the term containing $x^6$ is $3402$. 2. **Understand the expression:** The expression is $$px^2(5 + px)^8$$. 3. **Identify the term containing $x^6$:** The factor $px^2$ contributes $x^2$. So, from $(5 + px)^8$, we need the term containing $x^{6-2} = x^4$. 4. **General term in binomial expansion:** The general term in $(a + b)^n$ is $$\binom{n}{k} a^{n-k} b^k$$. Here, $a = 5$, $b = px$, $n = 8$. The term containing $x^4$ in $(5 + px)^8$ is when $k=4$: $$T_{k+1} = \binom{8}{4} 5^{8-4} (px)^4 = \binom{8}{4} 5^4 p^4 x^4$$ 5. **Multiply by $px^2$:** Multiplying by $px^2$ gives: $$p x^2 \times \binom{8}{4} 5^4 p^4 x^4 = \binom{8}{4} 5^4 p^{5} x^{6}$$ 6. **Coefficient of $x^6$ term:** The coefficient is: $$\binom{8}{4} 5^4 p^{5}$$ Given this coefficient equals $3402$: $$\binom{8}{4} 5^4 p^{5} = 3402$$ 7. **Calculate binomial coefficient and powers:** $$\binom{8}{4} = \frac{8!}{4!4!} = 70$$ $$5^4 = 625$$ So: $$70 \times 625 \times p^{5} = 3402$$ 8. **Simplify:** $$43750 p^{5} = 3402$$ 9. **Solve for $p^{5}$:** $$p^{5} = \frac{3402}{43750} = 0.07774$$ 10. **Find $p$:** $$p = \sqrt[5]{0.07774}$$ Using approximation: $$p \approx 0.57$$ **Final answer:** $$\boxed{p \approx 0.57}$$