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Essential Questions. How do we multiply polynomials? How do we use binomial expansion to expand binomial expressions that are raised to positive integer powers?.
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Essential Questions • How do we multiply polynomials? • How do we use binomial expansion to expand binomial expressions that are raised to positive integer powers?
Notice the coefficients of the variables in the final product of (a+ b)3. These coefficients are the numbers from the third row of Pascal's triangle. Each row of Pascal’s triangle gives the coefficients of the corresponding binomial expansion. The pattern in the table can be extended to apply to the expansion of any binomial of the form (a + b)n, where n is a whole number.
This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.
Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. 1. (k –5)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. [1(k)3(–5)0] + [3(k)2(–5)1] + [3(k)1(–5)2] + [1(k)0(–5)3] 2. (6m – 8)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. + [3(6m)1(–8)2] + [1(6m)0(–8)3] + [3(6m)2(–8)1] [1(6m)3(–8)0]
Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. 3. (x +2)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. + [3(x)2(2)1] + [3(x)1(2)2] + [1(x)0(2)3] [1(x)3(2)0] 4. (x –4)5 1 5 10 10 5 1 Identify the coefficients for n = 5, or row 5. + [10(x)2(–4)3] + [5(x)4(–4)1] + [10(x)3(–4)2] [1(x)5(–4)0] + [5(x)1(–4)4] + [1(x)0(–4)5]
Using Pascal’s Triangle to Expand Binomial Expressions Expand the expression. 5. (3x +1)4 1 4 6 4 1 Identify the coefficients for n = 4, or row 4. + [4(3x)3(1)1] [1(3x)4(1)0] + [6(3x)2(1)2] + [4(3x)1(1)3] + [1(3x)0(1)4]
