Question

factor this sum of cubes. 27x³ + 64 ( ?x + )( x² - 12x + ) hint: a³ ± b³ = (a ± b)(a² ∓ ab + b²)

Explanation:

Step1: Identify \(a\) and \(b\)

The sum of cubes formula is \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\). For \(27x^{3}+64\), we can rewrite \(27x^{3}\) as \((3x)^{3}\) and \(64\) as \(4^{3}\). So \(a = 3x\) and \(b = 4\).

Step2: Apply the sum of cubes formula

Using the formula \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\), substitute \(a = 3x\) and \(b = 4\).

• The first factor: \(a + b=3x + 4\), so the first box (coefficient of \(x\)) is \(3\), the second box is \(4\).
• The second factor: \(a^{2}-ab + b^{2}=(3x)^{2}-(3x)(4)+4^{2}=9x^{2}-12x + 16\). So the coefficient of \(x^{2}\) is \(9\) and the last box is \(16\).

Answer:

The filled - in factors are \((\boldsymbol{3}x + \boldsymbol{4})(\boldsymbol{9}x^{2}-12x+\boldsymbol{16})\)