Find a Particular Term in a Binomial Expansion
-$$\left( x ^ { 2 } + 3 \right) ^ { 9 }$$
A) $x ^ { 18 }+ 27 x ^{16} + 324 x ^ { 14 }$
B) $ ^ { 18 } + 27x ^{16} + 648 x ^ { 14 }$
C) $x ^ { 18 } + 30 x ^{16} + 324x ^ { 14 }$
D) $ x^ { 18 } + 30 x ^{16} + 648x ^ { 14 }$

Question

Find a Particular Term in a Binomial Expansion
-$$\left( x ^ { 2 } + 3 \right) ^ { 9 }$$
A) $x ^ { 18 }+ 27 x ^{16} + 324 x ^ { 14 }$
B) $ ^ { 18 } + 27x ^{16}  + 648 x ^ { 14 }$
C) $x ^ { 18 } + 30 x ^{16}  + 324x ^ { 14 }$
D) $ x^ { 18 } + 30 x ^{16}  + 648x ^ { 14 }$

Quizplus · Accepted Answer

The correct term can be found using the binomial theorem, which states that the $k$th term in the expansion of $(a + b)^n$ is given by $\binom{n}{k}a^{n-k}b^k$. For the term $x^{16}$, $k = 1$ since we are decreasing the power of $x^2$ by one step (from $x^{18}$ to $x^{16}$), so the coefficient is $\binom{9}{1}3^1 = 9 \times 3 = 27$. The term $x^{14}$ corresponds to $k = 2$, so its coefficient is $\binom{9}{2}3^2 = 36 \times 9 = 324$. Therefore, the correct expansion includes these terms as part of the sequence, matching option A.

Find a Particular Term in a Binomial Expansion - (x2+3)9\left( x ^ { 2 } + 3 \right) ^ { 9 }(x2+3)9

Find a Particular Term in a Binomial Expansion
- $\left( x ^ { 2 } + 3 \right) ^ { 9 }$