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Factor. \[ 27+8 x^{3} \]

Ask by Matthews Kirk. in the United States
Mar 14,2025

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Answer

The expression \(27+8x^3\) factors to \((3+2x)(9-6x+4x^{2})\).

Solution

Factor the expression \( 27+8x^3 \). Factor the expression by following steps: - step0: Factor: \(27+8x^{3}\) - step1: Rewrite the expression: \(3^{3}+\left(2x\right)^{3}\) - step2: Factor the expression: \(\left(3+2x\right)\left(3^{2}-3\times 2x+\left(2x\right)^{2}\right)\) - step3: Evaluate: \(\left(3+2x\right)\left(9-6x+4x^{2}\right)\) The factored form of the expression \(27+8x^3\) is \((3+2x)(9-6x+4x^{2})\).

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To factor \( 27 + 8x^3 \), we can recognize it as a sum of cubes. Specifically, \( 27 = 3^3 \) and \( 8x^3 = (2x)^3 \). The sum of cubes formula is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, let \( a = 3 \) and \( b = 2x \). Applying the formula: \[ 27 + 8x^3 = (3 + 2x)((3)^2 - (3)(2x) + (2x)^2) = (3 + 2x)(9 - 6x + 4x^2). \] Thus, the factored form is: \[ \boxed{(3 + 2x)(9 - 6x + 4x^2)}. \]

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