What do you do with a remainder in synthetic division?

The remainder in synthetic division represents the value left over after dividing a polynomial by a linear factor. You can express this remainder in one of two ways, as stated in the reference: either as a fraction or using the letter "R".

Expressing the Remainder

Here's a breakdown of how to handle the remainder:

• As a Fraction: The remainder becomes the numerator of a fraction, and the divisor (the linear term you divided by) becomes the denominator.

  • For example, if you divide a polynomial by x - 2 and get a remainder of 3, you'd express the remainder as 3/(x - 2). The complete quotient would then be written as the polynomial quotient plus 3/(x-2).

• Using "R": Write "R" followed by the remainder value.

  • For example, if the remainder is 5, you would simply write R 5 after the polynomial quotient.

Example

Let's say, after performing synthetic division, you have the following results:

• Polynomial being divided: x^2 + 3x + 5
• Divisor: x - 1
• Quotient from synthetic division: x + 4
• Remainder: 9

Then, the complete answer can be expressed as:

• As a Fraction: x + 4 + 9/(x - 1)
• Using "R": x + 4 R 9

Summary Table

Method	Description	Example (Remainder = 7, Divisor = x+3)
As a Fraction	Remainder as numerator, divisor as denominator.	7/(x+3)
Using "R"	"R" followed by the remainder.	R 7