Synthetic Division and Linear Factors

Review: Synthetic Division Synthetic division is a quick way of dividing a polynomial by a binomial to test for a zero. To synthetically divide, follow this process: Make three rows. Write the test zero and the coefficients of your polynomial in the first row. Drop the first coefficient to the third row. Multiply your test zero by the leftmost term in the third row. Place this product in the second row, one column over and add it to the coefficient in that column, placing the result in the third row. Repeat the last two steps until you have “used up” every coefficient. The last term in the third row is your remainder, the rest are the coefficients of your quotient.

Using Synthetic Division to Find Linear Factors It’s possible to use synthetic division to find linear factors of a polynomial. If, after synthetically dividing, your remainder is 0, your binomial (x minus the test zero) is a factor of your polynomial. This fact can be used to factor polynomials more quickly using synthetic division.

Example Factor x3-3x2-6x+8. Let’s start by listing possible rational zeroes of the polynomial using the rational root test. They are ±1, ±2, ±4, and ±8. Let’s see if 1 is a zero of the polynomial (and thus x-1 is a factor). Synthetically divide the polynomial by x-1:

Example (Cont.) Our remainder is 0, so x-1 is a factor of x3-3x2-6x+8. We can write the polynomial as (x-1)(x2 -2x -8). We can further factor our quadratic, leaving (x-1)(x-4)(x+2). This is our answer.

Try it Yourself Factor x3-9x2+26x-24 Factor x4+x3-3x2-21x-18 Factor x5+x4-19x3-49x2-30x

Answers (x-3)(x-2)(x-4) (x-3)(x^2 + 3x +6)(x+1) x(x+3)(x-5)(x+2)(x+1)