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 n choose k times a to the power of n minus k times b to the power of k.
Binomial coefficients are calculated using the formula n choose k 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 can also be found in Pascal's triangle.
For our expression (x + 3) cubed, we identify a equals x, b equals 3, and n equals 3. We substitute these values into the general binomial formula. This gives us the sum from k equals zero to 3 of 3 choose k times x to the power of 3 minus k times 3 to the power of k. When we write out all terms explicitly, we get four terms that we need to calculate.
Now we calculate each term separately. Term 1: 3 choose 0 times x cubed times 3 to the power of 0 equals 1 times x cubed times 1 which is x cubed. Term 2: 3 choose 1 times x squared times 3 to the power of 1 equals 3 times x squared times 3 which is 9 x squared. Term 3: 3 choose 2 times x times 3 squared equals 3 times x times 9 which is 27 x. Term 4: 3 choose 3 times 1 times 3 cubed equals 1 times 1 times 27 which is 27.
Combining all our calculated terms, we get the final expansion: (x + 3) cubed equals x cubed plus 9 x squared plus 27 x plus 27. The coefficients of this polynomial are 1, 9, 27, 27, which correspond to the powers of 3: 3 to the 0, 3 to the 1, 3 to the 2, and 3 to the 3. This pattern confirms our expansion is correct.
To verify our expansion is correct, we can substitute a specific value for x and check that both sides are equal. Let's use x equals 2. For the left side, we have (2 + 3) cubed which is 5 cubed equals 125. For the right side, we substitute into our expanded form: 2 cubed plus 9 times 2 squared plus 27 times 2 plus 27. This equals 8 plus 9 times 4 plus 54 plus 27, which is 8 plus 36 plus 54 plus 27 equals 125. Since both sides equal 125, our expansion is verified to be correct.