Definition: The binomial theorem is a formula used to expand binomial expressions raised to powers.
This theorem states that for any positive integer n:
Another method of expanding binomials involves Pascal’s triangle: the coefficients of the terms in the expansion (a + b)ⁿ correspond to the term in row n of Pascal’s triangle.
Example: Expand: (x + 4)³
Step 1: Identify the values to be plugged into the formula:
Since the exponent on our binomial is 3, we will plug n = 3 into the formula:
Step 2: Simplify:
Step 3: We can check this answer using Pascal’s triangle. As n = 3, we will look to the 3rd row of the triangle to find the coefficients of the binomial expansion:
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Checking back over our work, this verifies our calculations:
(x+4)³ = x³ + 12x² + 48x + 64
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