4.3 Division of Polynomials Divide polynomials by monomials Divide polynomials by polynomials Apply the division algorithm Learn synthetic division Understand the remainder theorem

Graphs of Polynomial Functions Example: Dividing by a monomial Solution

Example: Dividing polynomials (1 of 2) (5x³ − 4x² + 7x − 2) ÷ (x² + 1). Check your answer. Solution Quotient is 5x − 4 with a remainder of 2x + 2.

Example: Dividing polynomials (2 of 2) This result can also be written: Check:

Division Algorithm for Polynomials Let f(x) and d(x) be two polynomials, with degree of d(x) greater than zero and less than the degree of f(x). Then there exists unique polynomials q(x) and r(x) such that f(x) = d(x) · q(x) + r(x) Dividend = Divisor · Quotient + Remainder where either r(x) = 0 or the degree of r(x) is less than the degree of d(x). The polynomial r(x) is called the remainder.

Synthetic Division (1 of 2) A short cut called synthetic division can be used to divide x − k into a polynomial. Steps 1. Write k to the left and the coefficients of f(x) to the right in the top row. If any power does not appear in f(x), include a 0 for that term.

Synthetic Division (2 of 2) 2. Copy the leading coefficient of f(x) into the third row and multiply it by k. Write the result below the next coefficient of f(x) in the second row. Add the numbers in the second column and place the result in the third row. Repeat the process. 3. The last number in the third row is the remainder. If the remainder is 0, then the binomial x − k is a factor of f(x). The other numbers in the third row are the coefficients of the quotient, with terms written in descending powers.

Example: Performing synthetic division Use synthetic division to divide 2x³ + 4x² − x + 5 by x + 2. Solution Let k = −2 and perform the following.

Example: Finding the length of a rectangle If the area of a rectangle is x² + 8x + 15 and its width is x + 3, use division to find its length. Solution Use synthetic division to divide. remainder is 0 So, x + 3 divides evenly into x² + 8x + 15, and the length is x + 5.

Remainder Theorem If a polynomial f(x) is divided by x − k, the remainder is f(k). Thus f(k) is equal to the remainder in synthetic division. In a previous example f(−2) =2(−2)³ + 4(−2)² − (−2) + 5 = 7.