Austin H. answered • 07/23/14
Tutor
4.8
(6)
High School and Undergrad Level Math and Science Tutor
The original answer was too long so I will post the rest as a comment.
Long division:
For the example:
4x3-7x2+16x-5
———————
x2+2x-9
Write the equation as:
________________
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
Divide the first term of the dividend by the first term of the divisor and write the answer on the top of the line:
4x
________________
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
Multiply the answer by the divisor and subtract it from the dividend:
4x
________________
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
________________
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
- (4x3 + 8x2 - 36x)
————————
0 - 15x2 + 52x - 5
Repeat the previous two steps for the next term of the dividend:
4x - 7 ← quotient
________________
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
- (4x3 + 8x2 - 36x)
————————
-15x2 + 52x - 5
________________
x2 + 2x - 9 | 4x3 - 7x2 + 16x - 5
- (4x3 + 8x2 - 36x)
————————
-15x2 + 52x - 5
-(-7x2 - 14x + 63)
————————
-8x2 + 66x - 68 ← remainder
Repeat until the next term of the dividend is lower than the leading term of the divisor, in this case the next term of the dividend is 16x which is of lower order than x2
Finally, polynomial long division results in an answer of the form:
f(x) r(x)
––– = q(x) + –––
d(x) d(x)
where q(x) is the quotient and r(x) is the remainder, so the example above results in:
4x3 - 7x2 + 16x - 5 -8x2 + 66x + 58
————————— = (4x - 7) + ———————
x2 + 2x - 9 x2 + 2x - 9
————————— = (4x - 7) + ———————
x2 + 2x - 9 x2 + 2x - 9
To check your answer, multiply both sides by the divisor, multiply out, and simplify the right side:
4x3 - 7x2 + 16x - 5 = (x2 + 2x - 9)(4x - 7) + (-8x2 + 66x - 68)
= (4x3 + 8x2 - 36x - 7x2 - 14x + 63) + (-8x2 + 66x - 68)
= 4x3 - 7x2 + 16x - 5
I hope this explanation helps, but if you have any questions feel free to comment on this answer and I will reply.
Austin H.
—————————
x - 1
First, write the coefficients for the dividend inside an upside down division symbol leaving some room at the bottom, and all but the first coefficient of the divisor on the left:
1 | 1 4 -11 -30
|
————————
Take the first coefficient and drop it below the line:
1 | 1 4 -11 -30
|
————————
1
Multiply the dropped value by the left value and carry the result to the next column, above the line:
1 | 1 4 -11 -30
| 1
————————
1
Add the values in the next column and drop the result down:
1 | 1 4 -11 -30
| 1
————————
1 5
Repeat the previous two steps until you reach the last column:
1 | 1 4 -11 -30
| 1 5 -6
————————
1 5 -6 -36
The result is similar to long division, where 1, 5, and -6 are the coefficients of the quotient in decreasing order, and -36 is the remainder. The solution is written as:
x3 + 4x2 - 11x - 30 -36
———————— = (x2 + 5x - 6) + ———
x - 1 x - 1
As with long division, if the remainder is zero, then the divisor is a factor of the dividend. For this example the factors are (x - 3), (x + 5), and (x + 2) and the roots are 3, -5, -2:
3 | 1 4 -11 -30
| 3 21 30
————————
1 7 10 0
-5 | 1 4 -11 -30
| -5 5 30
————————
1 -1 -6 0
-2 | 1 4 -11 -30
| -2 -4 30
————————
1 2 -15 0
Synthetic division can also be used for polynomials where the divisor is of higher order than one, but the setup is slightly different:
Using the polynomial from the long division example:
4x3-7x2+16x-5
———————
x2+2x-9
Setup the synthetic division as before, but for the left hand side start with the last term of the divisor and place the coefficients one row left and one column down from the previous one, ignoring the first term (note the sign change on the left hand side) and drop the first coefficient of the dividend:
| 4 -7 16 -5
9 |
-2 |
————————
4
Next, multiply the dropped value by all the terms in the diagonal, and place them in a diagonal above the line, then add the values for the next column:
| 4 -7 16 -5
9 | 36 ← 4 * 9
-2 | -8 ← 4 * -2
————————
4 -15 ← -7 + -8
Repeat the previous step until the next diagonal would go beyond the last coefficient:
| 4 -7 16 -5
9 | 36 -135 ← -15 * 9
-2 | -8 30 ← -15 * -2
————————
4 -15 82
In this example, multiplying 82 by -2 will fit beneath the 5, but multiplying 82 by 9 will have to go off the right "edge" so we stop here.
Finally, add any remaining columns:
| 4 -7 16 -5
9 | 36 -135
-2 | -8 30
————————
4 -15 82 -140
Because there are two values to the left of the bar, the remainder is going to be degree 1, so the quotient is 4x - 15, and the remainder is 82x - 140, which gives the result:
4x3 - 7x2 + 16x - 5 82x - 140
————————— = (4x - 15) + ———————
x2 + 2x - 9 x2 + 2x - 9
This is a different answer than the one we got with long division, but checking the answer shows that it is also correct:
4x3 - 7x2 + 16x - 5 = (x2 + 2x - 9)(4x - 15) + (82x - 140)
= (4x3 + 8x2 - 36x - 15x2 - 30x + 135) + (82x - 140)
= 4x3 - 7x2 + 16x - 5
07/23/14