2x^4-3x^3+x+1 divided by 2x^2+x+1

Question

2x⁴-3x³+x+1 divided by 2x²+x+1 Long division and remainder

Answer 1

Long division in algebra is done the same way it's done in arithmetic, but you have to be careful to handle the variables properly.  For instance, in the problem you gave, the first expression has no x² term, so you have to insert 0x² as a placeholder, making it 

 

2x⁴ - 3x³ + 0x² +1

 

1) you write your divisor,  2x² + x + 1 to the left of the first expression.

 

2) since the first term of your expression (2x⁴) divided by the first term of your divisor (2x²) is x², the first term of your dividend will be x².  Multiplying this by the divisor will give  2x⁴ + x³ + x², which you write under the first expression and then subtract (some people prefer to change the sign of each term that product, making it -2x⁴ -x³ - x² and then add it, as it is less confusing to add). Either way, the result is  -4x³ - x².  

 

3) You then bring down the next term (+x) from the first expression, giving you -4x³ -x² +x, and repeat step 2. The next term of your dividend will be -2x, and the product of that term and the divisor will be -4x³ - 2x² -2x, which you write under the -4x³ -x² +x and subtract again, giving x² + 3x, and bring down the last term of the first expression (+1), giving you x² + 3x + 1, and it's back to step 2.

 

4) dividing x² by 2x² gives us ½, the next term in your dividend. You multiply ½ by our divisor, getting x² + ½x + ½. Subtracting this gives a remainder of 2½x + ½, which you add to your dividend as (2.5x + 0.5 ÷ 2x² + x + 1).