4.2 – Synthetic Division

Just as with real numbers, we can use division in regards to polynomials Two different methods; we will focus on what is known as Synthetic Division

Division Alogorithm If p(x) and d(x) are polynomials, and d(x) has a lesser degree (or equal) to p(x), then there exists polynomials q(x) and r(x) such that: p(x) = q(x) * d(x) + r(x) – r(x) is the remainder

If we divide two polynomials, then there will exist a new polynomial, and possibly a remainder To help facitilate faster division, we can use division similar to “long” division

Synthetic Division Synthetic division can be used to divide the polynomial f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 by x – k To use synthetic division, we do the following: – Pull out all coefficients of powers – Fill in missing powers with 0 – Use a simple algorithm – Rewrite new polynomial with powers and coefficients still left

A polynomial is considered a zero, if and only if, the remainder is zero Otherwise, not a zero Think of as a “speed-factoring” Algorithm: Drag down, multiply, add to next column, repeat

Example. Determine if the given k is zero of the following polynomial. If not, find p(k) p(x) = -2x x 3 – 5x 2 – 3x + 15; k = 5 (x – 5) – List out all coefficients of the powers in order

Pull out the remaining coefficients; this is now your “factored” polynomial with the x – 5 (x – k).

Example. Determine if the given k is zero of the following polynomial. If not, find p(k). p(x) = x 4 – 1; k = 1

Example. Determine if the given k is zero of the following polynomial. If not, find p(k). p(x) = x 2 – 6x + 13

Assignment Pg odd