The binomial theorem provides a formula to expand expressions of the form (a + b) to the power of n. The general formula is (a + b) to the n equals the sum from k equals zero to n of the binomial coefficient n choose k times a to the power of n minus k times b to the power of k. This allows us to systematically expand binomial expressions.
Binomial coefficients can be calculated using factorials. For n choose k, it equals n factorial divided by k factorial times n minus k factorial. For our expansion with n equals 3, we need the coefficients 3 choose 0, 3 choose 1, 3 choose 2, and 3 choose 3. These values are 1, 3, 3, and 1 respectively. These same values can also be found in Pascal's triangle.
To expand (x + 2) cubed, we identify the components: a equals x, b equals 2, and n equals 3. We then apply the binomial theorem formula. This gives us four terms: 3 choose 0 times x cubed times 2 to the 0, plus 3 choose 1 times x squared times 2 to the 1, plus 3 choose 2 times x to the 1 times 2 squared, plus 3 choose 3 times x to the 0 times 2 cubed.
Now we calculate each term individually. Term 1: 3 choose 0 is 1, x cubed is x cubed, and 2 to the 0 is 1, so we get x cubed. Term 2: 3 choose 1 is 3, x squared is x squared, and 2 to the 1 is 2, so we get 6x squared. Term 3: 3 choose 2 is 3, x to the 1 is x, and 2 squared is 4, so we get 12x. Term 4: 3 choose 3 is 1, x to the 0 is 1, and 2 cubed is 8, so we get 8.
Combining all our calculated terms, we get the final expanded form: (x + 2) cubed equals x cubed plus 6x squared plus 12x plus 8. This expansion follows the pattern of the binomial theorem, where each term has the correct binomial coefficient and powers of x and 2.
To verify our expansion is correct, let's substitute x equals 1 into both the original expression and our expanded form. For the original expression: (1 + 2) cubed equals 3 cubed which equals 27. For our expanded form: 1 cubed plus 6 times 1 squared plus 12 times 1 plus 8 equals 1 plus 6 plus 12 plus 8 which also equals 27. Since both expressions equal 27, our expansion is verified.