CK-12-Pre-Calculus Concepts

2.3. Polynomial Expansion and Pascal’s Triangle http://www.ck12.org

1

( 1

2 x

) 5

+ 5

( 1

2 x

) 4

(− 3 )+ 10

( 1

2 x

) 3

(− 3 )^2 + 10

( 1

2 x

) 2

(− 3 )^3 + 5

( 1

2 x

)

(− 3 )^4 + 1 ·(− 3 )^5

=x

5

32 −

15 x^4

16 +

90 x^3

8 −

270 x^2

4 +

405 x

2 −^243

Remember to simplify fractions.

=x

5

32 −

15 x^4

16 +

45 x^3

4 −

135 x^2

2 +

405 x

2 −^243

Concept Problem Revisited
Pascal’s triangle allows you to identify that the coefficients of( 2 x+ 3 )^5 will be 1, 5, 10, 10, 5, 1 like in Example
C. By carefully substituting, the expansion will be:

1 ·( 2 x)^5 + 5 ·( 2 x)^4 · 3 + 10 ·( 2 x)^3 · 32 + 10 ·( 2 x^2 )· 33 + 5 ( 2 x)^1 · 34 + 35

Simplifying is a matter of arithmetic, but most of the work is done thanks to the patterns of Pascal’s Triangle.

Vocabulary

Abinomial expansionis a polynomial that can be factored as the power of a binomial.
Pascal’s Triangleis a triangular array of numbers that describes the coefficients in a binomial expansion.

Guided Practice

1. Factor the following polynomial by recognizing the coefficients.
   x^4 + 4 x^3 + 6 x^2 + 4 x+ 1


2. Factor the following polynomial by recognizing the coefficients.
   8 x^3 − 12 x^2 + 6 x− 1


3. Expand the following binomial using Pascal’s Triangle.
   (A−B)^6
   Answers:
   1.(x+ 1 )^4


4. Notice that the first term of the binomial must be 2x, the last term must be -1 and the power must be 3. Now all
   that remains is to check.
   ( 2 x− 1 )^3 = ( 2 x)^3 + 3 ( 2 x)^2 ·(− 1 )+ 3 ( 2 x)^1 (− 1 )^2 +(− 1 )^3 = 8 x^3 − 12 x^2 + 6 x− 1
   3.(A−B)^6 =A^6 − 6 A^5 B+ 15 A^4 B^2 − 20 A^3 B^3 + 15 A^2 B^4 − 6 AB^5 +B^6