Polynomial synthetic division examples

Polynomial synthetic division

Synthetic division is a simplified and fast method for dividing a polynomial by a linear divisor of the type  x−c. It eliminates the requirement for lengthy division, making the operation faster and simpler, particularly for higher-degree polynomials. Here’s an explanation of how Polynomial synthetic division works, followed by an illustration.

Steps for Synthetic Division

Given a polynomial

​

Set up the coefficients: Record the coefficients of the polynomial P(x) in a row. If any powers of  x are absent (e.g., no 𝑥 2  phrase), include a zero for that term.

Write the root of the divisor: The divisor $x-c$ means the root is $c$. This is the value you’ll use for the division.

Perform synthetic division:

• Bring down the first coefficient as it is.
• Multiply the result by 𝑐 and add it to the following coefficient.
• Repeat the method for each coefficient.

Interpret the results:

• The last number in the row is the remainder.
• The other numbers are the quotient’s coefficients.

Polynomial synthetic division examples: Divide

Step-by-step process

1. Write the coefficients: The polynomial 2x^3 – 3x^2 + 4x – 5 has the coefficients $2,-3,4,-5$.

2. Set up the synthetic division table:To divide by x−1, use 𝑐 = 1 .

3. Begin the synthetic division:

• Bring down the first coefficient (2).
• Multiply 2 by 1 (the root of the divisor x – 1) to obtain 2.
• Add this to the following coefficient:− 3 + 2 = − 1
• Multiplying -1 by 1 yields -1.
• Add this to the following coefficient: 4 + ( − 1 ) = 3 .
• Multiply 3 by 1 to get three.
• Add this to the final coefficient: −5 + 3 = −2

The synthetic division table now looks like this:

4. Interpret the results:

The quotient is  2x^ 2 -x+3.
The remaining is -2.

Thus, the division gives:

Important Points:
Quotient: The quotient’s coefficients are  2x^2 −x+3.
remaining:  -2 is the remaining.
Interpretation: This indicates that P(x)=(x−1)(2x^ 2 −x+3)−2, indicating that the division was correct with the exception of −2.

Why It Works:

Synthetic division uses the remainder theorem and polynomial properties to divide by a linear term ( x – c). Rather than going through the long division procedure step by step, synthetic division integrates all of the processes into a more compact form.