B. In the expansion of $$ (3 x-2)^{10} $$, what is the: i. $$ 6^{\mathrm{\omega}} $$ term ii. coefficient of the $$ 4^{\dagger ext { term }} $$ term iii. third term in ascending powers of $$ x $$

Question

B. In the expansion of $$ (3 x-2)^{10} $$, what is the: i. $$ 6^{\mathrm{\omega}} $$ term ii. coefficient of the $$ 4^{\dagger 	ext { term }} $$ term iii. third term in ascending powers of $$ x $$

UpStudy AI · Accepted Answer

To find the $$6^{\text{th}}$$ term in the expansion of $$(3x-2)^{10}$$, we can use the binomial theorem. The binomial theorem states that for any positive integer $$n$$, the expansion of $$(a+b)^n$$ is given by:

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

In this case, $$a = 3x$$ and $$b = -2$$. The $$6^{\text{th}}$$ term in the expansion corresponds to the term where $$k = 5$$, since the terms are indexed from 0.

Therefore, the $$6^{\text{th}}$$ term in the expansion of $$(3x-2)^{10}$$ is:

$$
\binom{10}{5} (3x)^{10-5} (-2)^5
$$

Now, let's calculate the coefficient of the $$4^{\text{th}}$$ term in the expansion of $$(3x-2)^{10}$$. The coefficient of the $$4^{\text{th}}$$ term corresponds to the term where $$k = 3$$.

The coefficient of the $$4^{\text{th}}$$ term is given by:

$$
\binom{10}{3} (3x)^{10-3} (-2)^3
$$

Finally, to find the third term in ascending powers of $$x$$, we need to find the term where $$k = 2$$.

The third term in ascending powers of $$x$$ is given by:

$$
\binom{10}{2} (3x)^{10-2} (-2)^2
$$

Let's calculate these values.
Simplify the expression by following steps:
- step0: Solution:
  $$10cho\times ose\times 5\left(3x\right)^{5}\left(-2\right)^{5}$$
- step1: Multiply the terms:
  $$50cho\times ose\left(3x\right)^{5}\left(-2\right)^{5}$$
- step2: Multiply the terms:
  $$50cho^{2}se\left(3x\right)^{5}\left(-2\right)^{5}$$
- step3: Multiply the numbers:
  $$50echo^{2}s\left(3x\right)^{5}\left(-2\right)^{5}$$
- step4: Multiply the terms:
  $$12150echo^{2}sx^{5}\left(-2\right)^{5}$$
- step5: Multiply the numbers:
  $$-388800echo^{2}sx^{5}$$
Calculate or simplify the expression $$ 10 choose 3 * (3x)^7 * (-2)^3 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$10cho\times ose\times 3\left(3x\right)^{7}\left(-2\right)^{3}$$
- step1: Multiply the terms:
  $$30cho\times ose\left(3x\right)^{7}\left(-2\right)^{3}$$
- step2: Multiply the terms:
  $$30cho^{2}se\left(3x\right)^{7}\left(-2\right)^{3}$$
- step3: Multiply the numbers:
  $$30echo^{2}s\left(3x\right)^{7}\left(-2\right)^{3}$$
- step4: Multiply the terms:
  $$65610echo^{2}sx^{7}\left(-2\right)^{3}$$
- step5: Multiply the numbers:
  $$-524880echo^{2}sx^{7}$$
Calculate or simplify the expression $$ 10 choose 2 * (3x)^8 * (-2)^2 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$10cho\times ose\times 2\left(3x\right)^{8}\left(-2\right)^{2}$$
- step1: Multiply the terms:
  $$20cho\times ose\left(3x\right)^{8}\left(-2\right)^{2}$$
- step2: Multiply the terms:
  $$20cho^{2}se\left(3x\right)^{8}\left(-2\right)^{2}$$
- step3: Multiply the numbers:
  $$20echo^{2}s\left(3x\right)^{8}\left(-2\right)^{2}$$
- step4: Multiply the terms:
  $$131220echo^{2}sx^{8}\left(-2\right)^{2}$$
- step5: Multiply the numbers:
  $$524880echo^{2}sx^{8}$$
The $$6^{\text{th}}$$ term in the expansion of $$(3x-2)^{10}$$ is $$-388800x^5$$.

The coefficient of the $$4^{\text{th}}$$ term in the expansion of $$(3x-2)^{10}$$ is $$-524880x^7$$.

The third term in ascending powers of $$x$$ is $$524880x^8$$.
The $$6^{\text{th}}$$ term in the expansion of $$(3x-2)^{10}$$ is $$-388800x^5$$. The coefficient of the $$4^{\text{th}}$$ term is $$-524880x^7$$, and the third term in ascending powers of $$x$$ is $$524880x^8$$.

B. In the expansion of , what is the:
i. term
ii. coefficient of the term
iii. third term in ascending powers of

Upstudy AI Solution

Answer

Solution

Extra Insights

Part i: The term

Part ii: Coefficient of the term

Part iii: The third term in ascending powers of

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