multiplication by use of detached coefficient

Multiply using the method of detached coefficient

Multiply (1 – x +2x² +x³) by (1 +x- 2x³ + x⁴) as far as 4^th power of x.

In using this method the coefficients must be arranged in ascending  power of x. 0 as coefficient must be used for missing powers of x.

x⁰            x¹            x²            x³                  x⁴            x⁵                   ...... x⁷

1              -1            2              1                              .. 

1              1              0              -2            1              ..

1              -1            2              1              0              ..  (multiplying 1with all the above numbers)

                1             -1             2              1              ..  (multiplying 1with all the above numbers)

                                               -2             2              ..   (multiplying -2 with all the above numbers)

                                                               1              ..    (multiplying 1 with all the above numbers)

1              0              1              1             4              are the coefficients of rising power of x from left to right

Note we have restricted all multiplications to x⁴.

So, the answer is 1 + x² + x³ +4x⁴

Multiply (1 +x +x² +x³ +x⁴)² as far as 4^th power of x.

In using this method the coefficients must be arranged in ascending power of x.

x⁰            x¹            x²            x³                  x⁴            x⁵                   ...... x⁸

1              1              1              1              1              ..

1              1              1              1              1              ..

1              1              1              1              1              ..  (multiplying 1with all the above numbers)

                1              1              1              1              ..  (multiplying 1with all the above numbers)

                                1              1              1              ..   (multiplying 1 with all the above numbers)

                                                1              1              ..    (multiplying 1 with all the above numbers)

                                                                1              --   (multiplying 1 with all the above numbers)

1              2              3              4              5              are the coefficients of rising power of x from left to right

Note we have restricted all multiplications to x⁴.

So, the answer is 1+ 2x +3x² + 4x³ +5x⁴