Find the quotient & remainder for (2x^4+4x^2+2)/(x^2-1) using long division or synthetic division and find degree of quotient.

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

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Divide `2x^4+4x^2+2` by `x^2-1` using synthetic division:

The coefficients of the dividend are 2 0 4 0 2 and the coefficients of the divisor are 1 0 -1.

Drop the first coefficient of the divisor and negate the other coefficients to get 0 1. The division:

     |2   0   4   0   2

  1 |          2   0   6

0   |     0   0   0
     -----------------

      2   0   6 | 0   8

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The solution is `2x^2+6+8/(2x^4+4x^2+2)` ; the degree of the quotient is 2.

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The procedure:

(1)"drop" the first coefficient of the dividend. (the 2)
(2) Multiply the 2 along the diagonal by 0 and 1
(3) vertically add the second column resulting in 0
(4) multiply the 0 along the diagonal by 0 and 1
(5) vertically add the third column resulting in 6
(6) multiply the 6 along the diagonal by 0 and 1
(7) vertically add column 4 resulting in 0
(8) since further multiplication along the diagonal goes beyond the last column, vertically add all remaining columns.
(9) There are 2 numbers to the left of the bar -- thus the remainder has degree 2-1=1. Starting at the right the remainder will be 0x+8
(10) The quotient will be the remaining numbers, increasing in degree from the right or `2x^2+0x+6` . Since there were three numbers after the remainder, the degree of the quotient is 2.

 

 

Sources:

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The polynomial (2x^4+4x^2+2) has to be divided by (x^2-1) using long division.

x^2 - 1 | 2x^4 + 4x^2 + 2 | 2x^2 + 6

..............2x^4 - 2x^2

-----------------------------------------

........................6x^2 + 2

........................6x^2 - 6

-----------------------------------------------

...................................8

When (2x^4+4x^2+2) is divided by (x^2-1) the quotient is 2x^2 + 6 and the remainder is 8. The degree of the quotient is 2.

theking00's profile pic

theking00 | Student, Kindergarten | (Level 3) eNoter

Posted on

Thanks for the different methods described by both the editors.

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