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Find the least common multiple of (x-3)^2, x^2-9, and (x+3)^3

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Answer:


\textsf{LCM}=(x+3)^3(x-3)^2

Explanation:


\boxed{\begin{minipage}{6cm}\underline{Difference of Two Squares Formula}\\\\$a^2-b^2=\left(a+b\right)\left(a-b\right)$\\ \end{minipage}}

Factor x² - 9 by applying the Difference of Two Squares formula:


\begin{aligned}\implies x^2-9&=x^2-3^2\\&=(x+3)(x-3)\end{aligned}

Rewrite each of the given expressions as an expansion of their factors:


\bullet \quad (x-3)^2=(x-3)(x-3)


\bullet \quad x^2-9=(x+3)(x-3)


\bullet \quad (x+3)^3=(x+3)(x+3)(x+3)

Therefore, we can see that each expression has a factor of (x - 3), (x + 3) or both (x - 3) and (x + 3).

The least common multiple (LCM) of a, b, and c is the smallest multiplier that is divisible by a, b and c.

Therefore, to find the LCM of the given expressions, multiply each factor with the highest power:


  • \textsf{LCM}=(x+3)^3(x-3)^2

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