How do you expand (x - 1)³ using binomial expansion?

The binomial theorem or binomial expansion expresses the algebraic expansion of powers of a binomial.

Answer: The value of (x - 1)³ is x³ - 3x² + 3x – 1.

Here is a comprehensive expansion to this binomial expression.

Explanation:

\((x+y)^a= \sum_{b=0}^a (a/b)x^{a-b} y^b \)

So, binomial expansion formula for (a + b)³ is,

⇒³\(C_{0}\) (a³b⁰) + ³\(C_{1}\) (a²b¹) + ³\(C_{2}\) (a¹b²) + ³\(C_{3}\) (a⁰b³)

By using the above formula we will expand (x - 1)³.

Here, a = x and b = (-1).

⇒ ³\(C_{0}\) x³ + ³\(C_{1}\) x² (-1)¹ + ³\(C_{2}\) x¹(-1)² + ³\(C_{3}\)(-1) ³

We know that, ³\(C_{0}\) = ³\(C_{3}\) = 1 and ³\(C_{1}\) = ³\(C_{2}\) = 3.

By substituting above values in the equation, we get,

⇒ 1 × x³ + 3 × x²(-1) + 3 × x(-1)² + 1 × (-1)³ 

⇒ x ³ - 3x ² + 3x - 1                                                                       

Thus, the value of (x - 1)³ is x³ - 3x² + 3x - 1.