Divide 6x4 + 2x3 – 6x2 – 14x – 1 by 3x + 1 using long division. Show all your work. Then
explain if 3x + 1 is a factor of the dividend.
Solution:
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
Divide 6x4 by 3x to get 2x3
Write 2x3 above the division sign.
Multiply 2x3 by 3x + 1 to get 6x4 + 2x3. Write this result under the polynomial
under the division sign, like this:
2x 3
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
Subtract 6x4 + 2x3 from 6x4 + 2x3, and write the result (0) beneath the bar at the
bottom. Then bring the -6x2 and -14x terms down, like this:
2x 3
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
0 − 6x 2 − 14x
Divide -6x2 by 3x to get -2x. Write this to the right of the 2x3 above the division
bar:
2x 3 − 2x
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
0 − 6x 2 − 14x
Multiply 3x + 1 by -2x to get -6x2 – 2x and write this result beneath the bottom
line:
2x 3 − 2x
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
0 − 6x 2 − 14x
− 6x 2 − 2x
Subtract -6x2 – 2x from -6x2 – 14x to get -12x. Write this below the bottom line
and bring the -1 term down as well:
2x 3 − 2x
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
0 − 6x 2 − 14x
− 6x 2 − 2x
− 12x − 1
Finally, divide -12x by 3x to get -4
Write this above the division sign, to the right of the -2x term:
2x 3 − 2x − 4
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
0 − 6x 2 − 14x
− 6x 2 − 2x
− 12x − 1
Multiply 3x + 1 by -4 to get -12x – 4. Write this below the bottom line, and
subtract:
2x 3 − 2x − 4
3x + 1 6x 4 + 2x 3 − 6x 2 − 14x − 1
6x 4 + 2x 3
0 − 6x 2 − 14x
− 6x 2 − 2x
− 12x − 1
−12x − 4
3
The “3” in the last line indicates that there is a remainder.
The solution is 2x 3 − 2x − 4 +
3
3x + 1
Since there is a remainder, 3x + 1 is not a factor of 6x4 + 2x3 – 6x2 – 14x – 1.