Recall the square binomial expansion (a+b)^2 = a^2+2ab+b^2.
See solution.
Practice makes perfect
To cube a binomial we can first rewrite it as the product of a first and a second power.
(a+b)^3 = (a+b)(a+b)^2
We can then expand the squared binomial, obtaining a perfect square trinomial. Finally we multiply the linear binomial left over with the obtained trinomial.
As we can see, this procedure is is not very efficient. However, we can identify a specific pattern in our result.
a^3+3a^2b+3ab^2+b^3
⇕
(1)a^3b^0+3a^2b^1+3a^1b^2+(1)a^0b^3
The exponents for the binomial's first term are decreasing by one, starting from 3 until reaching 0. Those for the other term are increasing by one, starting from 0 until reaching 3. Furthermore, the coefficients of the resulting terms coincide with the numbers from row 3 of Pascal's triangle.
cc
Row& Pascal's Triangle
c
0 1 2 3
&
cccccccccccc
& & & & & & 1 & & & & &
& & & & & 1 & & 1 & & & &
& & & & 1 & & 2 & & 1 & & &
& & & 1 & & 3 & & 3 & & 1 & &
We can use this pattern to cube a binomial in a more convenient way. Let's see an example. We will calculate (x+2)^3 by following this pattern.
(x+2)^3 & = (1)x^3(2)^0+3x^2(2)^1+3x^1(2)^2+(1)x^0(2)^3 [0.5em]
(x+2)^3 & = (1)x^3(1) +3x^2(2) +3x(4) +(1)(1)(8) [0.5em]
(x+2)^3 & = x^3+6x^2+12x+8
This is faster and easier than following the procedure that we did at first for a general case (a+b)^3.
