Algebra 2 Notes – Dividing Polynomials A. Long Division and Synthetic Division Division of Polynomials Given polynomials p x and d x 0 , there exists unique polynomials q x and r x such that p x d x q x r x Where r x 0 or r x is less than the degree of d x . d x is the divisor, q x is the quotient, and r x is the remainder. Dividing a polynomial by a polynomial using long division: Comparison between dividing integers and dividing polynomials Dividend Remainder Quotient Divisor Divisor Compute 579 ÷ 16 Compute (5x2 + 7x +9) ÷ (x + 6) 1. 6 x3 7 x 2 6 x 6 2x 1 2. 8 9 x 2 x 2 12 x3 5 x5 x2 3 Algebra 2 Notes – Dividing Polynomials Dividing a polynomial by a binomial using synthetic division: THIS IS A SHORTCUT THAT ONLY WORKS WHEN THE DIVISOR IS A LINEAR BINOMIAL (I.E., THE DIVISOR IS x – c) !!! Comparison between long and synthetic division of polynomials Compute (2x – 3x2 – 4x + 11) ÷ (x – 2) using Compute (2x3 – 3x2 – 4x + 11) ÷ (x – 2) using long division. synthetic division. 3 2 x3 x 2 7 x 13 1. x2 f x 3. 2x3 x2 16x 15 ÷ (x – 2) x 4 8 x3 15 x 2 2 x 6 2. x3 f x 4. 3x3 2x 6 ÷ ( x 2 ) Algebra 2 Notes – Dividing Polynomials B. The Remainder Theorem The Remainder Theorem If a polynomial p x is divided by x c using synthetic division, then the remainder is p(c). Examples: a) b) x 4 2 x3 5 x 4 x 1 2 x3 3x 2 10 x2 c) f(x) = x3 + 2x2 - 5x + 7 f(1) = C. The Factor Theorem The Factor Theorem For a polynomial p x , p c 0 if and only if x c is a factor of p x . Examples: Determine if the following binomials are factors of f(x). a) ( x3 – 14x + 8) and (x + 4) b) (x4 – 6x3 – 40x + 33) and (x – 7)