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Use Pascal's Triangle to expand
(3z^(2)+5x^(2))^(4). Express your answer in simplest form.
Answer:

Use Pascal's Triangle to expand $\left(3 z^{2}+5 x^{2}\right)^{4}$ . 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(3 z^{2}+5 x^{2}\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 $5$ th row of Pascal's Triangle (since we start counting from the $0$ th row) is $1, 4, 6, 4, 1$ .
Write Coefficients: Write out each term of the expansion using the coefficients from Pascal's Triangle. $\newline$ The expansion will have $5$ terms, corresponding to the coefficients $1, 4, 6, 4, 1$ .
Apply Binomial Expansion: Apply the binomial expansion using the coefficients and the variables $(3z^{2})$ and $(5x^{2})$ . The terms will be: $1\cdot(3z^{2})^{4}\cdot(5x^{2})^{0} + 4\cdot(3z^{2})^{3}\cdot(5x^{2})^{1} + 6\cdot(3z^{2})^{2}\cdot(5x^{2})^{2} + 4\cdot(3z^{2})^{1}\cdot(5x^{2})^{3} + 1\cdot(3z^{2})^{0}\cdot(5x^{2})^{4}$
Calculate Each Term: Calculate each term separately. $\newline$ $1$ st term: $1*(3z^{2})^4*(5x^{2})^0 = 1*(81z^{8})*(1) = 81z^{8}$ $\newline$ $2$ nd term: $4*(3z^{2})^3*(5x^{2})^1 = 4*(27z^{6})*(5x^{2}) = 540z^{6}x^{2}$ $\newline$ $3$ rd term: $6*(3z^{2})^2*(5x^{2})^2 = 6*(9z^{4})*(25x^{4}) = 1350z^{4}x^{4}$ $\newline$ $4$ th term: $4*(3z^{2})^1*(5x^{2})^3 = 4*(3z^{2})*(125x^{6}) = 1500z^{2}x^{6}$ $\newline$ $5$ th term: $1*(3z^{2})^0*(5x^{2})^4 = 1*(1)*(625x^{8}) = 625x^{8}$
Combine Final Form: Combine all the terms to get the final expanded form. $\newline$ The expanded form is: $\newline$ $81z^{8} + 540z^{6}x^{2} + 1350z^{4}x^{4} + 1500z^{2}x^{6} + 625x^{8}$

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