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

Use Pascal's Triangle to expand $\left(x^{2}+2 z\right)^{4}$ . 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(x^{2}+2 z\right)^{4}

. Express your answer in simplest form.

\newline

Answer:

Identify Row: Identify the row of Pascal's Triangle that corresponds to the exponent $4$ . The fourth row (starting with row $0$ ) of Pascal's Triangle is $1, 4, 6, 4, 1$ .
Write Expansion Terms: Write out the terms of the expansion using the coefficients from Pascal's Triangle. $\newline$ The expansion will have terms that correspond to the coefficients $1, 4, 6, 4, 1$ . Each term will be of the form $(x^{2})^{4-k} \times (2z)^{k}$ , where $k$ is the term number starting from $0$ .
Calculate Each Term: Calculate each term of the expansion. $\newline$ The terms are: $\newline$ $1$ st term: $(x^{2})^{4-0} \times (2z)^{0} = x^{8} \times 1 = x^{8}$ $\newline$ $2$ nd term: $4 \times (x^{2})^{4-1} \times (2z)^{1} = 4 \times x^{6} \times 2z = 8x^{6}z$ $\newline$ $3$ rd term: $6 \times (x^{2})^{4-2} \times (2z)^{2} = 6 \times x^{4} \times (2z)^{2} = 6 \times x^{4} \times 4z^{2} = 24x^{4}z^{2}$ $\newline$ $4$ th term: $4 \times (x^{2})^{4-3} \times (2z)^{3} = 4 \times x^{2} \times (2z)^{3} = 4 \times x^{2} \times 8z^{3} = 32x^{2}z^{3}$ $\newline$ $5$ th term: $(x^{2})^{4-4} \times (2z)^{4} = 1 \times (2z)^{4} = 16z^{4}$
Combine Terms: Combine all the terms to write the expanded form of the polynomial. $\newline$ The expanded form is: $\newline$ $x^{8} + 8x^{6}z + 24x^{4}z^{2} + 32x^{2}z^{3} + 16z^{4}$

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