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Use Pascal's Triangle to expand
(3z^(2)-4)^(3). Express your answer in simplest form.
Answer:

Use Pascal's Triangle to expand $\left(3 z^{2}-4\right)^{3}$ . Express your answer in simplest form. $\newline$ Answer:

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Pascal's triangle and the Binomial Theorem

Full solution

Q. Use Pascal's Triangle to expand

\left(3 z^{2}-4\right)^{3}

. Express your answer in simplest form.

\newline

Answer:

Identify Row: Identify the row of Pascal's Triangle that corresponds to the exponent of the binomial expansion. $\newline$ Since we are expanding $(3z^{2}-4)^{3}$ , we need the fourth row of Pascal's Triangle (since the rows are indexed starting from $0$ ). The fourth row of Pascal's Triangle is $1, 3, 3, 1$ .
Write Expansion Terms: Write out each term of the expansion using the coefficients from Pascal's Triangle. $\newline$ The expansion will have four terms, corresponding to the coefficients $1, 3, 3, 1$ . Each term will be of the form $(\text{coefficient}) \times (\text{first term in the binomial raised to a power}) \times (\text{second term in the binomial raised to a power})$ . The powers on the first term will decrease from $3$ to $0$ , and the powers on the second term will increase from $0$ to $3$ .
Calculate First Term: Calculate the first term of the expansion. $\newline$ Using the first coefficient from Pascal's Triangle, which is $1$ , the first term is $(1) \times (3z^{2})^{3} \times (-4)^{0}$ . This simplifies to $(1) \times (27z^{6}) \times (1)$ , which is just $27z^{6}$ .
Calculate Second Term: Calculate the second term of the expansion. $\newline$ Using the second coefficient from Pascal's Triangle, which is $3$ , the second term is $(3) \times (3z^{2})^{2} \times (-4)^{1}$ . This simplifies to $(3) \times (9z^{4}) \times (-4)$ , which is $-108z^{4}$ .
Calculate Third Term: Calculate the third term of the expansion. $\newline$ Using the third coefficient from Pascal's Triangle, which is $3$ , the third term is $(3) \times (3z^{2})^{1} \times (-4)^{2}$ . This simplifies to $(3) \times (3z^{2}) \times (16)$ , which is $144z^{2}$ .
Calculate Fourth Term: Calculate the fourth term of the expansion. $\newline$ Using the fourth coefficient from Pascal's Triangle, which is $1$ , the fourth term is $(1) \times (3z^{2})^{0} \times (-4)^{3}$ . This simplifies to $(1) \times (1) \times (-64)$ , which is $-64$ .
Combine Terms: Combine all the terms to write the final expanded form. $\newline$ The expanded form of $(3z^{2}-4)^{3}$ is $27z^{6} - 108z^{4} + 144z^{2} - 64$ .

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