Binomial Theorem

Definition: The binomial theorem is a formula used to expand binomial expressions raised to powers.

This theorem states that for any positive integer n:

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Where:

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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

Solution:

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:gif.latex-%5Cbg_white %28x+4%29%5E3%3D%5Csum%5E3_%7Bk%3D0%7D %5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D 3%5C%5C k %5Cend%7Barray%7D%5Cright%29x%5E%7B3-k%7Dy%5Ek%5C%5C%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bc%7D 3%5C%5C 0 %5Cend%7Barray%7D%5Cright%29x%5E%7B.gif

Step 2: Simplify:

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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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