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Synthetic Division with Integers Solutions

Synthetic division is a shorthand for dividing a polynomial dividend by a linear divisor.

Here are the steps for solving with synthetic division:

1. 1. Set the linear divisor equal to zero and solve.

1. 1. Place the zero of the linear divisor in the top left corner.

1. 1. List the coefficients and constant of the polynomial dividend (remembering to use 0 for any missing degree).

1. 1. Bring down the first coefficient under the line.

1. 1. Multiply the zero of the linear divisor by the first coefficient and place it under the second coefficient.

1. 1. Add the column vertically.

1. 1. Repeat until no values are left.

1. 1. The value in the bottom right corner represents the remainder . If the remainder is zero, then the linear divisor is a factor of the polynomial.

1. 1. Write the solution as:

Example 1

Simplify using synthetic division.
$(5 x^{4} + 12 x^{3} - 6 x^{2} - 14 x - 8) \div (x + 2)$

Implement

$\begin{array}{rcl} x + 2 & = & 0 \\ x & = & - 2 \end{array}$

$5 x^{3} + 2 x^{2} - 10 x + 6 - \frac{20}{x + 2}$

Explain

Set the linear divisor equal to zero and solve

Note

This is the root or zero of the divisor.

Place the zero of the linear divisor in the top left corner
List the coefficients and constant of the polynomial dividend
Write the first coefficient under the line
Multiply the constant r by the first coefficient and place it under the second coefficient
Add the column vertically
Repeat until no values are left
Determine if there is a remainder or if the divisor is a factor
Write the quotient with the remainder

Note

When writing the quotient, start with one less degree than the polynomial dividend. From there, the degree of each term decreases by one.

Example 2

Simplify using synthetic division.

$\frac{9 x^{3} - 36 x^{2} - 4 x + 16}{x - 4}$ $\begin{array}{rcl} x - 4 & = & 0 \\ x & = & 4 \end{array}$

$9 x^{2} - 4$ or $(3 x - 2) (3 x + 2)$

Note

For solutions that have no remainder, check if the quotient can be factored. The rationale behind factoring the quotient completely will be covered in a later lesson. For example, the quotient of this example is also a difference of two squares and would be written as the factored expression (3x – 2)(3x + 2).

Remember to write the final expression in standard form, factoring completely when possible.

Example 3

Find the quotient using synthetic division.

$(3 p^{3} - 8 p + 14) \div (p - 6)$ $\begin{array}{rcl} p - 6 & = & 0 \\ p & = & 6 \end{array}$

$3 p^{2} + 18 p + 100 + \frac{614}{p - 6}$