Skill: 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

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

by

$$x-2.$$

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:

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