Synthetic Division is a process that allows us to quickly divide a polynomial by a linear binomial: (x - a). It is worth noting that our divisor must be able to be written in the format of (x - a). Where x is our variable with a coefficient of 1 and is raised to the power of 1. Additionally, a can be any real number.

Test Objectives
• Demonstrate an understanding of polynomial long division
• Demonstrate the ability to find a quotient using synthetic division
Synthetic Division Test:

#1:

Instructions: Find each quotient.

$$a)\hspace{.2em}\frac{x^3 + 9x^2 + 26x + 30}{x + 5}$$

$$b)\hspace{.2em}\frac{x^3 + 15x^2 + 60x + 28}{x + 7}$$

#2:

Instructions: Find each quotient.

$$a)\hspace{.2em}\frac{7x^3 + 51x^2 + 50x - 24}{x + 6}$$

$$b)\hspace{.2em}\frac{3x^3 + 2x^2 - 29x - 24}{x + 3}$$

#3:

Instructions: Find each quotient.

$$a)\hspace{.2em}\frac{x^4 - 6x^3 + 4x^2 - x + 25}{x - 5}$$

$$b)\hspace{.2em}\frac{x^4 + 2x^3 - 17x^2 + 11x - 12}{x - 3}$$

#4:

Instructions: Find each quotient.

$$a)\hspace{.2em}\frac{4x^4 - 3x^3 - 18x^2 + 20x - 6}{x - 2}$$

$$b)\hspace{.2em}\frac{x^4 - x^3 - 26x^2 + 23x + 27}{x - 5}$$

#5:

Instructions: Find each quotient.

$$a)\hspace{.2em}\frac{x^4 + 9x^3 + 15x^2 - 5x - 9}{x + 2}$$

$$b)\hspace{.2em}\frac{4x^3 - 2x + x^4 - 15}{x + 4}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x^2 + 4x + 6$$

$$b)\hspace{.2em}x^2 + 8x + 4$$

#2:

Solutions:

$$a)\hspace{.2em}7x^2 + 9x - 4$$

$$b)\hspace{.2em}3x^2 - 7x - 8$$

#3:

Solutions:

$$a)\hspace{.2em}x^3 - x^2 - x - 6 - \frac{5}{x - 5}$$

$$b)\hspace{.2em}x^3 + 5x^2 - 2x + 5 + \frac{3}{x - 3}$$

#4:

Solutions:

$$a)\hspace{.2em}4x^3 + 5x^2 - 8x + 4 + \frac{2}{x - 2}$$

$$b)\hspace{.2em}x^3 + 4x^2 - 6x - 7 - \frac{8}{x - 5}$$

#5:

Solutions:

$$a)\hspace{.2em}x^3 + 7x^2 + x - 7 + \frac{5}{x + 2}$$

$$b)\hspace{.2em}x^3 - 2 - \frac{7}{x + 4}$$