Binomial Theorem

Method to expand polynomials

The binomial theorem states a formula for expressing the powers of sums. The most succinct version of this formula is shown immediately below.

picture of binomial theorem as infinite series

Isaac Newton wrote a generalized form of the Binomial Theorem. However, for quite some time Pascal's Triangle had been well known as a way to expand binomials (Ironically enough, Pascal of the 17th century was not the first person to know about Pascal's triangle)

A closer look at the Binomial Theorem

The easiest way to understand the binomial theorem is to first just look at the pattern of polynomial expansions below.

  • (x+y)² =x²+2xy + y²
  • (x+y)3 = x3 + 3x2Y+ 3xY2 + y3
  • (x+y)4=x4+ 4x3Y +6x2Y2 + 4XY3 + Y4

Binomial Theorem Formula

The generalized formula for the pattern above is known as the binomial theorem
generalized example of biomial theorem

Practice Problems

on the Binomial Theorem

Problem 1

Use the formula for the binomial theorem to determine the fourth term in the expansion (y − 1)7

Problem 2

Make use of the binomial theorem formula to determine the eleventh term in the expansion (2a − 2)12

Problem 3

Use the binomial theorem formula to determine the fourth term in the expansion

$$ _7 C _3 (3x)^{7-3} \left( -\frac{2}{3}\right)^3 \\ 35 (3x)^4 \cdot \frac{-8}{27} \\\ 35 \cdot 3^3 \cdot 3x^4 \cdot \frac{-8}{27} \\ 35 \cdot 27 \cdot 3 x^4 \cdot \frac{-8}{27} \\ 35 \cdot \cancel{\color{red}{27}} 3x^4 \cdot \frac{-8}{ \cancel{\color{red}{27}} } \\ \boxed{-840 x^4} $$