Question

factor this difference of cubes. 27x³ - 8 (\boxed{?}x + \boxed{})(\boxed{}x² + 6x + \boxed{}) hint: a³ ± b³ = (a ± b)(a² ∓ ab + b²)

Explanation:

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

The formula for the difference of cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\). For \(27x^3-8\), we can rewrite \(27x^3=(3x)^3\) and \(8 = 2^3\). So \(a = 3x\) and \(b = 2\).

Step2: Apply the difference of cubes formula

Substitute \(a = 3x\) and \(b = 2\) into the formula \((a - b)(a^2+ab + b^2)\). First, \(a - b=3x - 2\). Then, \(a^2=(3x)^2 = 9x^2\), \(ab=(3x)\times2 = 6x\), and \(b^2=2^2 = 4\). So the factored form is \((3x - 2)(9x^2+6x + 4)\). Comparing with the given form \((\square x+\square)(\square x^2 + 6x+\square)\), the first square (coefficient of \(x\)) is \(3\), the second square (constant term in the first binomial) is \(- 2\) (but since the form has a plus sign, we note the formula's structure: the first binomial is \(a - b\), so in the given form \((\square x+\square)\), it's \(3x+(- 2)\)), the third square (coefficient of \(x^2\)) is \(9\), and the fourth square (constant term in the trinomial) is \(4\).

Answer:

The filled - in form is \((\boldsymbol{3}x+\boldsymbol{-2})(\boldsymbol{9}x^2 + 6x+\boldsymbol{4})\) (or if we consider the sign in the binomial as per the formula's application, the factored form is \((3x - 2)(9x^2+6x + 4)\) and the blanks are filled with \(3\), \(-2\), \(9\), \(4\) respectively).