Consider the expansion of $$ (x-3)^{8} $$. (a) Write down the number of terms in this expansion. (b) Find the coefficient of the term in $$ x^{6} $$.

Question

UpStudy AI · Accepted Answer

To address the expansion of $$ (x - 3)^8 $$, we'll tackle each part of the problem step by step.

### (a) Number of Terms in the Expansion

When expanding a binomial expression of the form $$ (x + a)^n $$, the number of distinct terms in the expansion is always $$ n + 1 $$. This is because each term corresponds to a different power of $$ x $$, ranging from $$ x^n $$ down to $$ x^0 $$.

For the given expression $$ (x - 3)^8 $$:

- **Exponent $$ n $$**: 8
- **Number of terms**: $$ 8 + 1 = 9 $$

**Answer to (a):** There are **9 terms** in the expansion.

### (b) Coefficient of the $$ x^6 $$ Term

To find the coefficient of $$ x^6 $$ in the expansion of $$ (x - 3)^8 $$, we can use the Binomial Theorem, which states:

$$
(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k
$$

For $$ (x - 3)^8 $$, $$ a = -3 $$ and $$ n = 8 $$. We're interested in the term where the power of $$ x $$ is 6, which means:

$$
8 - k = 6 \quad \Rightarrow \quad k = 2
$$

Now, compute the coefficient using $$ k = 2 $$:

$$
\text{Coefficient} = \binom{8}{2} \cdot (-3)^2
$$

Calculate each part:

1. **Binomial Coefficient**:
   $$
   \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28
   $$
   
2. **Exponentiation**:
   $$
   (-3)^2 = 9
   $$

3. **Final Coefficient**:
   $$
   28 \times 9 = 252
   $$

**Answer to (b):** The coefficient of the $$ x^6 $$ term is **252**.

### Summary

- **(a)** There are **9 terms** in the expansion of $$ (x - 3)^8 $$.
- **(b)** The coefficient of the $$ x^6 $$ term is **252**.
There are 9 terms in the expansion of $$ (x - 3)^8 $$, and the coefficient of the $$ x^6 $$ term is 252.

Consider the expansion of .
(a) Write down the number of terms in this expansion.
(b) Find the coefficient of the term in .

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions

Consider the expansion of . (a) Write down the number of terms in this expansion. (b) Find the coefficient of the term in .