Use Pascal's Triangle to expand (4-z^(2))^(4). Express your answer in simplest form. 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: Write out the terms of the expansion using the coefficients from Pascal's Triangle. The expansion will have the form: 1*(4)^4*(-z^2)^0 + 4*(4)^3*(-z^2)^1 + 6*(4)^2*(-z^2)^2 + 4*(4)^1*(-z^2)^3 + 1*(4)^0*(-z^2)^4.Simplify Terms: Simplify each term of the expansion. 1*(4)^4*(-z^2)^0 = 1*256*1 = 256 4*(4)^3*(-z^2)^1 = 4*64*(-z^2) = -1024z^2 6*(4)^2*(-z^2)^2 = 6*16*z^4 = 96z^4 4*(4)^1*(-z^2)^3 = 4*4*(-z^6) = -64z^6 1*(4)^0*(-z^2)^4 = 1*1*z^8 = z^8Combine Simplified Terms: Combine all the simplified terms to get the final expanded form. The final expanded form is: 256 - 1024z^2 + 96z^4 - 64z^6 + z^8.

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Pascal's triangle and the Binomial Theorem

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Q. Use Pascal's Triangle to expand

\left(4-z^{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 fourth row (starting with row $0$ ) of Pascal's Triangle is $1, 4, 6, 4, 1$ .
Write Expansion: Write out the terms of the expansion using the coefficients from Pascal's Triangle. $\newline$ The expansion will have the form: $\newline$ $1\times(4)^4\times(-z^2)^0 + 4\times(4)^3\times(-z^2)^1 + 6\times(4)^2\times(-z^2)^2 + 4\times(4)^1\times(-z^2)^3 + 1\times(4)^0\times(-z^2)^4$ .
Simplify Terms: Simplify each term of the expansion. $\newline$ $1\times(4)^4\times(-z^2)^0 = 1\times256\times1 = 256$ $\newline$ $4\times(4)^3\times(-z^2)^1 = 4\times64\times(-z^2) = -1024z^2$ $\newline$ $6\times(4)^2\times(-z^2)^2 = 6\times16\times z^4 = 96z^4$ $\newline$ $4\times(4)^1\times(-z^2)^3 = 4\times4\times(-z^6) = -64z^6$ $\newline$ $1\times(4)^0\times(-z^2)^4 = 1\times1\times z^8 = z^8$
Combine Simplified Terms: Combine all the simplified terms to get the final expanded form. $\newline$ The final expanded form is: $\newline$ $256 - 1024z^2 + 96z^4 - 64z^6 + z^8$ .