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) = anxn + an-1xn-1 + … + a1x + a0 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) = -2x4 + 11x3 – 5x2 – 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) = x4 – 1; k = 1

• Example. Determine if the given k is zero of the following polynomial. If not, find p(k).

• p(x) = x2 – 6x + 13

• Assignment• Pg. 320• 19-37 odd