binomial Expansion using Pascal's triangle?

Question

How do I use Pascal's Triangle to expand these two binomials?

 

(x-6) ^ 6

(2x -3) ^ 4
 

Please explain the process if possible. Thanks.

 

Answer 1

 1 )   ( x - 6 ) ⁶

Pascals triangle : - 

     

                                       1

                   1         1

              1        2       1

         1      3        3      1

      1     4        6        4       1 

   1    5      10     10       5       1  

1    6     15    20      15      6       1

( x - 6 ) ⁶ = x ⁶  + 6 x ⁵  ( - 6 ) +   15 x ⁴ ( - 6 ) ²+ 20 x ³ ( - 6 ) ³  +   15 x ² ( - 6 ) ⁴ +6 x ( - 6 )⁵    + ( - 6 )⁶ 

( x - 6 ) ⁶ = x ⁶  - 36 x ⁵   +   15x ⁴ ( 36 )  -  20 x ³ ( 6 ) ³  +   15 x ² ( 6 ) ⁴ -  6 x (  6 )⁵    + (  6 )⁶

( x - 6 ) ⁶  =  x ⁶  - 36 x ⁵   + 540 x ⁴   -  4320 x ³ +   19440x ²  -  46656 x + 46656.

 2 )

 ( 2 x - 3 ) ⁴                                                

   Pascals triangle : -
          

                   1         1

              1        2       1

         1      3        3      1

      1     4        6        4       1 

 

( 2 x - 3 ) ⁴ =( 2 x )⁴  + 4 (2 x)³  ( - 3) +   6 (2 x )² ( - 3) ²+ 4 (2 x ) +(- 3 )³   + 1 *(- 3 )⁴

( 2 x - 3 ) ⁴ = 16 x⁴  - 96 x³  + 21 6 x²  -  216 x + 81.

Answer 2

The following expanded powers of (a + b)ⁿ, where a + b is any binomial and n is a whole number. Look for patterns.

Each expansion is a polynomial. There are some patterns to be noted.

1. There is one more term than the power of the exponent, n. That is, there are terms in the expansion of (a + b)ⁿ.

2. In each term, the sum of the exponents is n, the power to which the binomial is raised.

3. The exponents of a start with n, the power of the binomial, and decrease to 0. The last term has no factor of a. The first term has no factor of b, so powers of b start with 0 and increase to n.

4. The coefficients start at 1 and increase through certain values about "half"-way and then decrease through these same values back to 1.

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

Pascals triangle : - (x + y)ⁿ = (x - 6)⁶.

n=0----->                               1

n=1----->                         1         1

n=2----->                   1         2       1

n=3----->              1         3        3         1

n=4----->          1       4         6        4         1 

n=5----->      1      5       10       10       5       1  

n=6----->  1     6      15        20      15       6       1

(x - 6)⁶ = x⁶ + 6x⁵(- 6) + 15x⁴(- 6)² + 20x³(- 6)³ + 15x² (- 6)⁴ + 6x(- 6)⁵ + (- 6)⁶

(x - 6)⁶ = x⁶ - 36x⁵ + 15x⁴(36) - 20x³(6)³ + 15x²(6)⁴ - 6x(6)⁵ + (6)⁶

(x - 6)⁶ = x⁶ - 36x ⁵ + 540x⁴ - 4320x³ +19440x² - 46656x + 46656.

2 )(2x - 3)⁴.

Pascals triangle : -  (x + y)ⁿ = (2x - 3)⁴.

n=0----->                               1

n=1----->                         1         1

n=2----->                   1         2       1

n=3----->              1         3        3         1

n=4----->          1       4         6        4         1

 

(2x - 3)⁴ = (2x )⁴ + 4 (2x)³(- 3) + 6(2x )² (- 3) ²+ 4(2 x ) + (- 3 )³ + 1(- 3 )⁴

(2x - 3)⁴ = 16 x⁴ - 96 x³ + 21 6x² - 216x + 81.