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Find the binomial that completes the factorization. $\newline$ $u^3 + 27 = (\underline{\hspace{3cm}})(u^2 - 3u + 9)$

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Factor sums and differences of cubes

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Q. Find the binomial that completes the factorization.

\newline

u^3 + 27 = (\underline{\hspace{3cm}})(u^2 - 3u + 9)

Recognize Cubes: Recognize the expression $u^3 + 27$ as a sum of cubes, where $u^3$ is a cube and $27$ is $3^3$ , which is also a cube.
Use Sum of Cubes Formula: Use the sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ . Here, $a$ is $u$ and $b$ is $3$ .
Write Binomial: Write down the binomial $(u + 3)$ that represents the sum of $u$ and $3$ .
Check Formula with Multiplication: Check the formula by multiplying the binomial $(u + 3)$ with the trinomial $(u^2 - 3u + 9)$ to see if it equals $u^3 + 27$ .
Perform Multiplication: Perform the multiplication: u + \(3)(u^ $2$ - $3$ u + $9$ ) = u^ $3$ - $3$ u^ $2$ + $9$ u + $3$ u^ $2$ - $9$ u + $27$ \
Combine Like Terms: Combine like terms: $u^3 + 27$ . This confirms that the binomial $(u + 3)$ is the correct factor.

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