Divide polynomials using synthetic division

Synthetic division represents division of a polynomial by a linear divisor $x-c$ through a structured sequence of multiply-and-add operations on the polynomial’s coefficients, including zero coefficients for missing powers. It yields the quotient and remainder, with the final value equal to $P(c)$ , thereby connecting polynomial division to the Remainder and Factor Theorems; this treatment is limited to linear divisors in the form $x-c$ , not generalized synthetic division for higher-degree or nonmonic divisors.

Detailed Explanation: Divide polynomials using synthetic division

To divide a polynomial by a linear divisor of the form $(x-c)$ , use $c$ in the synthetic division process.

Suppose we want to divide

P(x)=2x^3-5x+6

x-2.

1. Write the coefficients

The polynomial has no $(x^2)$ -term, so include a $0$ for that missing coefficient:

2x^3+0x^2-5x+6

The coefficients are $(2,0,-5,6)$ . Since the divisor is $(x-2)$ , use $2$ :

2. Bring down the first coefficient

Bring down the $2$ :

3. Multiply and add repeatedly

Multiply the $2$ by the number outside, $2$ , and write the result under the next coefficient:

2\cdot 2=4

Add:

0+4=4

Continue multiplying and adding:

The final row is $(2,4,3,12)$ .

4. Interpret the result

The last number is the remainder:

\text{remainder}=12.

The other numbers are the coefficients of the quotient:

2x^2+4x+3.

Therefore,

\boxed{\frac{2x^3-5x+6}{x-2} =2x^2+4x+3+\frac{12}{x-2}}

or, using division form,

\boxed{2x^3-5x+6=(x-2)(2x^2+4x+3)+12}.

The remainder $(12)$ also equals $(P(2))$ , illustrating the Remainder Theorem:

P(2)=2(2)^3-5(2)+6=12.

Learn by doing: Divide polynomials using synthetic division

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Synthetic Division - Divide with a Remainder

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