(x^4-10x^2+2x+3)/(x-3) using long division

Question

I would like to see the work to this.

Answer 1

x ^4 - 10 x ^2 + 2 x + 3 can be divided by x - 3.

Given that

To solve this equation in long division we have to write it as

x - 3 | x  ^4 - 10 x ^2 + 2 x  +  3 | x ^3 + 3 x ^2  -  x

          x ^4 - 3 x ^3

( - )     -----------------------------------        ( change symbols )

                 3 x ^3 - 10 x ^2 + 2 x  

                  3 x ^3 - 9 x ^2

  ( - )    --------------------------------------       ( change symbols )

                            - x ^2 + 2 x  + 3

                          - x ^2  + 3 x

 ( - )   --------------------------------------         ( change symbols )

                                   - x + 3

Therefore the reminder of the given equation  ( - x + 3 ).

Answer 2

Rewrite the expression in long division form (x⁴ - 10x² + 2x + 3)/(x - 3).

Because there is no x³ - term in the dividend, we  need to line up the subtraction by using zero coefficients (or leaving spaces) for the missing terms.

Divide the first term of the dividend by the first term of the divisor x⁴/x = x³.

So,the first term of the quotient is x³. Multiply (x - 3) by x³ and subtract.

Divide the first term of the last row by first term of the divisor 3x³/x = 3x².

So,the second term of the quotient is 3x². Multiply (x - 3) by 3x² and subtract.

Divide the first term of the last row by first term of the divisor  (- x²)/x = - x.

So,the third term of the quotient is (- x). Multiply (x - 3) by (- x) and subtract.

Divide the first term of the last row by first term of the divisor (- x)/x = - 1.

So,the fourth term of the quotient is - 1. Multiply (x - 3) by (- 1) and subtract.

The remainder is the last entry in the last row.There fore, R = 0.

The number along the bottom row are the coefficients of the quotient.

The result of the division is (x⁴ - 10x² + 2x + 3)/(x - 3) = (x³ + 3x² - x - 1) + 0/(x - 3).

The result of the division is x⁴ - 10x² + 2x + 3 = (x - 3)(x³ + 3x² - x - 1) + 0.