IXL - Pascal's triangle and the Binomial Theorem (Precalculus practice)

Use Pascal's Triangle to complete the expansion of

(

)

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Learn with an example

question

Use Pascal's Triangle to complete the expansion of

(

)

key idea

Pascal's Triangle is a triangular array of numbers. Row

of Pascal's Triangle lists the coefficients for the expanded binomial

(

)

The top row is row 0, and only consists of 1 since

(

)

Here are the first few rows of Pascal's Triangle.

Row 0,

(

)

Row 1,

(

)

Row 2,

(

)

Row 3,

(

)

Row 4,

(

)

Here are some facts about Pascal's Triangle:

The first and last number in each row is 1.
All other numbers are greater than 1 and can be found by adding the closest two numbers in the previous row.
For example, the second number in row 2 is 2 because the two numbers closest to it in the previous row (1 and 1) add to 2.

Each row in the triangle is symmetric.

To expand

(

)

using Pascal's Triangle, write the terms

–

etc. in descending order of degree on

and ascending order of degree on

with the coefficients in row

For example, to expand

(

)

using Pascal's Triangle, write the terms

and

in descending order of degree on

and ascending order of degree on

with the coefficients in row 2: 1, 2, and 1.

(

)

solution

Use Pascal's Triangle to expand

(

)

In this case,

and

So, look at the coefficients in row 3.

Row 0,

(

)

Row 1,

(

)

Row 2,

(

)

Row 3,

(

)

Write the terms in descending order of degree on

and ascending order of degree on

with the coefficients in row 3: 1, 3, 3, and 1.

(

)

Identify the missing coefficient.

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