Use synthetic division to divide the polynomial 2x^3 – 12x – 5 by x + 4,
Write the quotient polynomial and the remainder. [Be careful – notice that there is no x2 term.]. Show work.
Synthetic division is the same as long polynomial division, except that we write down only the coefficients, and not the complete terms. For this to be valid, any missing term must be compensated by a 0 (read the note [Be careful...]).
Synthetic division is very similar to numeric long division that we are used to, except each digit is replaced by a number, positive or negative. The layout is almost the same.
You can read up
http://en.wikipedia.org/wiki/Polynomial_long_division
for an example.
The work is not difficult, but difficult to do a nice layout by posting.
Try it out and see if you have difficulties.
The quotient is 2x³-8x+20 with a remainder of -85.
erratum:
The quotient is 2x²-8x+20 with a remainder of -85.
76
To perform synthetic division, follow these steps:
Step 1: Write down the coefficients of the polynomial in descending order of powers of x:
Polynomial: 2x^3 – 12x – 5
Coefficients: 2, 0, -12, -5
Step 2: Set up the synthetic division grid. Place the divisor, x + 4, on the left side of the grid.
-4
Step 3: Bring down the first coefficient (2) from the polynomial into the empty box below the line of the grid.
-4 | 2 0 -12 -5
2
Step 4: Multiply the divisor, -4, by the number in the bottom box (which is 2 in this case). Write the result below the next coefficient, and add it to the next coefficient.
-4 | 2 0 -12 -5
2
______________
-8
Next coefficient: -12 + (-8) = -20
Step 5: Repeat this process for all the coefficients, bringing down the next coefficient and performing the multiplication and addition.
-4 | 2 0 -12 -5
2 -20
______________
-8 -32
Next coefficient: -32 + (-32) = -64
Step 6: Continue this process until you have performed all the necessary multiplications and additions.
-4 | 2 0 -12 -5
2 -20 64
______________
-8 -32 -69
The final row represents the coefficients of the quotient polynomial: 2x - 20. The final number, -69, is the remainder.
Therefore, the quotient polynomial is 2x - 20, and the remainder is -69.