To divide the polynomial
2x3−x2−x+1 by
x2+2x, we can use polynomial long division. Here are the steps:
1. **Setup the Division**:
2x3−x2−x+1x2+2x
2. **Divide the Leading Terms**:
Divide the leading term of the numerator
2x3 by the leading term of the denominator
x2:
2x3x2=2x
3. **Multiply and Subtract**:
Multiply
2x by the entire divisor
x2+2x:
2x⋅(x2+2x)=2x3+4x2
Subtract this result from the original polynomial:
(2x3−x2−x+1)−(2x3+4x2)=−5x2−x+1
4. **Repeat the Process**:
Now, divide the new leading term
−5x2 by the leading term of the divisor
x2:
−5x2x2=−5
Multiply
−5 by the entire divisor
x2+2x:
−5⋅(x2+2x)=−5x2−10x
Subtract this result from the current polynomial:
(−5x2−x+1)−(−5x2−10x)=9x+1
5. **Write the Final Result**:
The quotient is
2x−5 and the remainder is
9x+1. So, the division can be expressed as:
2x3−x2−x+1x2+2x=2x−5+9x+1x2+2x
Therefore, the result of the division is:
2x−5+9x+1x2+2x