Question

factor this sum of cubes.
$x^3 + 1,000$
$(x + ?)(x^2 + x + )$
hint: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

Explanation:

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

Given \(x^3 + 1000\), we know \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). Here, \(a = x\) (since \(x^3=a^3\)) and \(b^3 = 1000\), so \(b=\sqrt[3]{1000}=10\).

Step2: Apply the sum of cubes formula

Using the formula \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\) with \(a = x\) and \(b = 10\), we substitute:

• The first factor is \((x + b)=(x + 10)\).
• The second factor: \(a^2=x^2\), \(-ab=-x\times10=-10x\), \(b^2 = 10^2 = 100\), so the second factor is \((x^2-10x + 100)\).

Answer:

The factored form is \((x + 10)(x^2-10x + 100)\), so the values in the boxes are \(10\), \(- 10\), and \(100\) respectively.