Question

3a) factor $1 - 125n^3$
b) choose the polynomial identity you used in 3a).
a) square of a binomial
b) difference of two squares
c) sum of two cubes
d) difference of two cubes

Explanation:

3a)

Step1: Identify the form

We have \(1 - 125n^{3}\), which can be written as \(1^{3}-(5n)^{3}\) since \(125 = 5^{3}\) and \((5n)^{3}=125n^{3}\).

Step2: Apply the difference of cubes formula

The formula for the difference of two cubes is \(a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})\). Here, \(a = 1\) and \(b = 5n\).
Substituting these values into the formula, we get:
\((1-(5n))(1^{2}+(1)(5n)+(5n)^{2})=(1 - 5n)(1 + 5n+25n^{2})\)

The expression \(1-125n^{3}\) is in the form of \(a^{3}-b^{3}\) (where \(a = 1\) and \(b = 5n\)), which is the form for the difference of two cubes. The square of a binomial is \((a\pm b)^{2}=a^{2}\pm2ab + b^{2}\), the difference of two squares is \(a^{2}-b^{2}=(a - b)(a + b)\), and the sum of two cubes is \(a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})\). So the correct identity is the difference of two cubes.

Answer:

\((1 - 5n)(1 + 5n + 25n^{2})\)

3b)