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Use Pascal's Triangle to expand (2-5y)^(4). Express your answer in simplest form. Answer:

Use Pascal's Triangle to expand (2−5y)4 (2-5 y)^{4} . Express your answer in simplest form.\newlineAnswer:

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Full solution

Q. Use Pascal's Triangle to expand (2−5y)4 (2-5 y)^{4} . Express your answer in simplest form.\newlineAnswer:
  1. Identify Row: Identify the row of Pascal's Triangle that corresponds to the exponent 44. The row for the exponent 44 is the fifth row (starting with row 00 for the exponent 00), which is 1,4,6,4,11, 4, 6, 4, 1.
  2. Write Terms: Write out the terms using the binomial coefficients from Pascal's Triangle.\newlineThe terms will be:\newline1×(2)4×(−5y)0+4×(2)3×(−5y)1+6×(2)2×(−5y)2+4×(2)1×(−5y)3+1×(2)0×(−5y)41\times(2)^4\times(-5y)^0 + 4\times(2)^3\times(-5y)^1 + 6\times(2)^2\times(-5y)^2 + 4\times(2)^1\times(-5y)^3 + 1\times(2)^0\times(-5y)^4
  3. Simplify Terms: Simplify each term. \newline1×(16)×1+4×(8)×(−5y)+6×(4)×(−5y)2+4×(2)×(−5y)3+1×1×(−5y)4=16−160y+6×4×25y2−8×125y3+625y41\times(16)\times1 + 4\times(8)\times(-5y) + 6\times(4)\times(-5y)^2 + 4\times(2)\times(-5y)^3 + 1\times1\times(-5y)^4 = 16 - 160y + 6\times4\times25y^2 - 8\times125y^3 + 625y^4
  4. Perform Multiplications: Continue simplifying by performing the multiplications.\newline=16−160y+600y2−1000y3+625y4= 16 - 160y + 600y^2 - 1000y^3 + 625y^4
  5. Final Expression: Write the final expression in standard polynomial form, with the terms in descending order of the exponent.\newline=625y4−1000y3+600y2−160y+16= 625y^4 - 1000y^3 + 600y^2 - 160y + 16

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