Use Pascal's Triangle to expand (3-5x^(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 of Pascal's Triangle (starting with row 0) is 1, 4, 6, 4, 1.Write Expansion Terms: Write out the terms of the expansion using the coefficients from Pascal's Triangle. The expansion will have the form: (3-5x^2)^4 = 1*(3^4)*(-5x^2)^0 + 4*(3^3)*(-5x^2)^1 + 6*(3^2)*(-5x^2)^2 + 4*(3^1)*(-5x^2)^3 + 1*(3^0)*(-5x^2)^4Calculate Each Term: Calculate each term of the expansion. 1st term: 1*(3^4)*(-5x^2)^0 = 1*81*1 = 81 2nd term: 4*(3^3)*(-5x^2)^1 = 4*27*(-5x^2) = -540x^2 3rd term: 6*(3^2)*(-5x^2)^2 = 6*9*25x^4 = 1350x^4 4th term: 4*(3^1)*(-5x^2)^3 = 4*3*(-125x^6) = -1500x^6 5th term: 1*(3^0)*(-5x^2)^4 = 1*1*625x^8 = 625x^8Combine Final Form: Combine all the terms to write the final expanded form. (3-5x^2)^4 = 81 - 540x^2 + 1350x^4 - 1500x^6 + 625x^8

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Pascal's triangle and the Binomial Theorem

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Q. Use Pascal's Triangle to expand

\left(3-5 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 fourth row of Pascal's Triangle (starting with row $0$ ) 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 the form: $\newline$ $(3-5x^2)^4 = 1\cdot(3^4)\cdot(-5x^2)^0 + 4\cdot(3^3)\cdot(-5x^2)^1 + 6\cdot(3^2)\cdot(-5x^2)^2 + 4\cdot(3^1)\cdot(-5x^2)^3 + 1\cdot(3^0)\cdot(-5x^2)^4$
Calculate Each Term: Calculate each term of the expansion. $\newline$ $1$ st term: $1*(3^4)*(-5x^2)^0 = 1*81*1 = 81$ $\newline$ $2$ nd term: $4*(3^3)*(-5x^2)^1 = 4*27*(-5x^2) = -540x^2$ $\newline$ $3$ rd term: $6*(3^2)*(-5x^2)^2 = 6*9*25x^4 = 1350x^4$ $\newline$ $4$ th term: $4*(3^1)*(-5x^2)^3 = 4*3*(-125x^6) = -1500x^6$ $\newline$ $5$ th term: $1*(3^0)*(-5x^2)^4 = 1*1*625x^8 = 625x^8$
Combine Final Form: Combine all the terms to write the final expanded form. $\newline$ $(3-5x^2)^4 = 81 - 540x^2 + 1350x^4 - 1500x^6 + 625x^8$