College Algebra & Trigonometry, 2018a

2.6. POLYNOMIAL LONG DIVISION 119
A = B ∗ Q + R
2x 4 +7x 3 +4x 2 − 2x − 1=(x 2 +3x +1)∗ (2x 2 +1x − 1) + 0
=2x 4 +6x 3 +2x 2
= 1x 3 +3x 2 + x
= −1x 2 − 3x − 1
=2x 4 +7x 3 +5x 2 − 2x − 1
Now we also know that the remainder is zero, because x 2 +3x +1divides evenly
into 2x 4 +7x 3 +4x 2 − 2x − 1 and so:
2x 4 +7x 3 +4x 2 − 2x − 1=(x 2 +3x +1)∗ (2x 2 +1x − 1)
This method makes the reasoning behind dividing polynomials somewhat more
apparent than the long division process, but it is more cumbersome. The way
that polynomial long division is usually approached is as follows:
x 2 +3x +1 ) 2x 4 +7x 3 +4x 2 − 2x − 1
Then, just as we did in the other method, we question “What should we multiply
x 2 by to get 2x 4 ?” Answer: “2x 2 ” This is the first term in our answer:
2x 2
x 2 +3x +1 ) 2x 4 +7x 3 +4x 2 − 2x − 1
Then we multiply 2x 2 (x 2 +3x +1)and change all the signs to see what we’ll be
left with:
2x 2
x 2 +3x +1 ) 2x 4 +7x 3 +4x 2 − 2x − 1
− 2x 4 − 6x 3 − 2x 2