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

Use Pascal's Triangle to expand $\left(5 y^{2}+x^{2}\right)^{3}$ . 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

\left(5 y^{2}+x^{2}\right)^{3}

. Express your answer in simplest form.

\newline

Answer:

Identify Row: Identify the row of Pascal's Triangle that corresponds to the exponent of the binomial expansion. $\newline$ Since we are expanding $(5y^{2}+x^{2})^{3}$ , we need the $4^{\text{th}}$ row of Pascal's Triangle, which corresponds to the coefficients for a cubic expansion. $\newline$ The $4^{\text{th}}$ row of Pascal's Triangle is $1, 3, 3, 1$ .
Write Terms: Write out each term of the expansion using the binomial theorem and the coefficients from Pascal's Triangle. $\newline$ The binomial theorem tells us that $(a+b)^n = \Sigma (\binom{n}{k}) \cdot a^{(n-k)} \cdot b^k$ , where $\Sigma$ denotes the sum over $k$ from $0$ to $n$ . $\newline$ For $(5y^{2}+x^{2})^{3}$ , we have: $\newline$ $1\cdot(5y^{2})^3\cdot(x^{2})^0 + 3\cdot(5y^{2})^2\cdot(x^{2})^1 + 3\cdot(5y^{2})^1\cdot(x^{2})^2 + 1\cdot(5y^{2})^0\cdot(x^{2})^3$
Calculate Terms: Calculate each term of the expansion. $\newline$ Now we will calculate each term: $\newline$ $1\times(5y^{2})^{3}\times(x^{2})^{0} = 1\times(125y^{6})\times(1) = 125y^{6}$ $\newline$ $3\times(5y^{2})^{2}\times(x^{2})^{1} = 3\times(25y^{4})\times(x^{2}) = 75y^{4}x^{2}$ $\newline$ $3\times(5y^{2})^{1}\times(x^{2})^{2} = 3\times(5y^{2})\times(x^{4}) = 15y^{2}x^{4}$ $\newline$ $1\times(5y^{2})^{0}\times(x^{2})^{3} = 1\times(1)\times(x^{6}) = x^{6}$
Combine for Final Form: Combine all the terms to write the final expanded form. $\newline$ The final expanded form of $(5y^{2}+x^{2})^{3}$ is: $\newline$ $125y^{6} + 75y^{4}x^{2} + 15y^{2}x^{4} + x^{6}$

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