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

Use Pascal's Triangle to expand $(4+2 y)^{3}$ . 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

(4+2 y)^{3}

. Express your answer in simplest form.

\newline

Answer:

Identify Exponent $3$ : Identify the row of Pascal's Triangle that corresponds to the exponent $3$ . The third row of Pascal's Triangle (starting with row $0$ ) is $1, 3, 3, 1$ . These numbers will be the coefficients in the expanded form.
Write Binomial Theorem Terms: Write out the terms using the binomial theorem and the coefficients from Pascal's Triangle. $\newline$ The binomial theorem states that $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} \cdot a^{n-k} \cdot b^k$ . For $(4+2y)^3$ , the terms will be: $\newline$ $1\cdot(4)^3\cdot(2y)^0 + 3\cdot(4)^2\cdot(2y)^1 + 3\cdot(4)^1\cdot(2y)^2 + 1\cdot(4)^0\cdot(2y)^3$
Calculate Each Term: Calculate each term separately. $\newline$ $1\times(4)^3\times(2y)^0 = 1\times64\times1 = 64$ $\newline$ $3\times(4)^2\times(2y)^1 = 3\times16\times2y = 96y$ $\newline$ $3\times(4)^1\times(2y)^2 = 3\times4\times(2y)^2 = 3\times4\times4y^2 = 48y^2$ $\newline$ $1\times(4)^0\times(2y)^3 = 1\times1\times(2y)^3 = 8y^3$
Combine Terms: Combine all the terms to get the expanded form. $64 + 96y + 48y^2 + 8y^3$

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