Synthetic Division

Synthetic Division

What is synthetic division?

• Shortcut way to divide polynomials

• Uses the roots (not factor form) as the divisor

• Uses the coefficients of the terms, and not the variables, as the dividend

Steps to dividing by synthetic division

• 1. Change the factor form of the divisor to the root. (factor form is (x - 3), root is 3)

• 2. Write out the polynomial to be divided in descending form, writing in zeros for the terms that are missing. For example: x⁴ + 2x² + 4 would be written as: x⁴ + 0x³ + 2x² + 0x + 4

Steps to dividing by synthetic division (cont)

• 3. Write out the coefficients only of the polynomial, including signs and including the 0's

• 4. Write the root to the left of the coefficients with a vertical bar between it and the coefficients and a horizontal line underneath the coefficients (like an upside-down division symbol)

Steps to dividing by synthetic division (cont)

• 5. Bring the first coefficient down below the line, without doing anything to it

• 6. Multiply the coefficient by the root, and add the number to the next term.

• 7. Multiply the sum by the coefficient and add to the next term.

Steps to dividing by synthetic division (cont)

• Repeat this process until there are no more terms.

• If the remainder is 0, then the factor is a true root

• If the remainder is not 0, then the value of the remainder is the value of the polynomial evaluated at the x-value of the divisor (what you would get if you substituted the divisor into the polynomial)

Steps to dividing by synthetic division (cont)

• Note: You can only use this form if your divisor is a binomial

• Always remember to put in the missing terms with a zero coefficient

• The remaining number will start with a variable exponent of one less than the original. In the example to the right, the polynomial was 3rd degree, the quotient is 2nd degree.

Multiple Choice

x³-3x²-10x+24 ÷ x+3

• x²-7x+5
• x-8
• x²-6x+8
• x+9

Multiple Choice

(3x³ + 5x - 1) ÷ (x + 1)

• 3x³ - 3x² + 8x - 9
• 3x² - 3x + 8 - 9/x+1
• 3x² + 3x + 8 + 7/x+1
• 3x³ + 3x² + 8x + 7

Multiple Choice

(2x³ + 5x² + 9) ÷ (x + 3)

• 2x² - x + 3
• 2x³ - x² + 3x
• 2x² + 11x + 33 + 108/x+3
• 2x² - 5x + 12

Multiple Choice

Let f(x) = 4x² -33x -27.  Is k = 9 a zero of the function?

• yes
• no

Multiple Choice

Let f(x) = 3x² - 9x +2.
Is k = -3 a zero of the function?

• yes
• no

Multiple Choice

Cam divided (x⁴ + 3x² - 4x - 2) by factor of (x-2) using synthetic division. His work is shown above. Which best describes his mistake?

• Cam wrote the remainder incorrectly.

• Cam did not use a zero place holder for the x³ term.

• Cam added instead of subtracting the rows.

• Cam should have used -2 as his division since the factor was x-2.

Multiple Choice

What is the proper way to write the answer for the following problem?

• 2x³-x²-25x+12

• 2x⁴-x³-25x²-12x+0

• 2x³-x²-25x-12

• 2x⁴-x³-25x²-12x-0

Multiple Choice

Give the correct name for 
2x³+5x

• Linear Binomial
• Cubic Binomial
• Quadratic Trinomial
• Linear Trinomial

Multiple Choice

If you are missing an exponent in your polynomial you must put a zero as a coefficient place holder for that exponent

• True
• False

Multiple Choice

Which term is missing in this problem?
2x³ + 5x² + 9 ÷
x + 3

• x⁴
• x³
• x²
• x

Multiple Choice

What should be the order of the polynomial coefficients if
(3x - 4x³ + 6x⁴ + 1) / (x + 3)

• 3   0   -4   6   1
• 6   -4   3   1   0
• 6   -4   0   3   1
• -6   4   0   -3   -1

Multiple Choice

If you were dividing x⁶ + 4x³ + 2, how many 0's would you need when setting up the top row of your synthetic division?

• 0
• 2
• 4
• 6

Multiple Choice

For synthetic division, what would be the number in the left hand box?

• 0

• 1

• 2

• 3