Polynomial & Synthetic Division I. Performing Long Division on Polynomials. A) Divide as you did long division back in elementary school. 1) Work with the term with the biggest remaining degree. 2) Line up terms with the same degree. 3) Remember to put – (axn + …) before you collect like terms. B) When you divide, the answer is a factor of the original poly 1) Know as the Division Algorithm: f(x) = d(x)q(x) + r(x) a) d(x) is what you divided by (the divisor). b) q(x) is what you get after you divided (the quotient). c) r(x) is your remainder (if it didn’t factor perfectly). Example: (x2 – 3x – 4) ÷ (x + 1) = (x + 1) (x – 4) no remainder. f (x) = d(x) q(x) + r(x)

Polynomial & Synthetic Division examples: 2x2 + 6x + 4 x – 10 2x3 – 14x2 – 56x – 40 – (2x3 – 20x2) 6x2 – 56x – (6x2 – 60x) 4x – 40 – (4x – 40)

Polynomial & Synthetic Division examples: 3x3 + 2x2 – 4x + 1 x2 + 5 3x5 + 2x4 + 11x3 + 11x2 – 20x + 5 – (3x5 + 15x3) 2x4 – 4x3 + 11x2 – (2x4 + 10x2) – 4x3 + x2 – 20x – (– 4x3 – 20x) x2 + 5 – (x2 + 5)

Polynomial & Synthetic Division II.. Synthetic Substitution and Remainders. A) When you use synthetic substitution, the last number you get after performing the math is the remainder (r): the y coordinate for the specific value for “x” you tested. 1) Using synthetic substitution on k will give you the point (k , r) on the graph. Think (x , y). B) If the remainder is zero, then your “k” value is a solution to the polynomial (it is a root, an x-intercept, a solution, a zero, the answer, etc.). 1) You can factor out this value using synthetic division. C) Remainder theorem: If you test f(k) using syn substitution 1) (k , r) is a point on the graph. 2) If r = 0, then k is a root (solution) to f(x). An x-intercept. 3) If r = 0, then (x – k) is a factor of f(x). (Change sign)

Polynomial & Synthetic Division III.. Synthetic Division and Remainders. A To factor (x – k) from the original polynomial, change the sign of the “k” term and use synthetic substitution. 1) This is called Synthetic Division. B) The #s below the synthetic division bar are the coefficients of the new smaller polynomial you get after factoring by (x – k) 1) The smaller poly will have a degree (exp) one less than the original poly. 2) Try to factor the new smaller polynomial, using any factoring method until you cannot factor anymore. a) All factors will now say (x – k). b) Or if k is a fraction (k = b/a) then it will be (ax – b).

Polynomial & Synthetic Division Examples: (3x3 – 4x2 – 28x – 16) / (x + 2) – 2 3 – 4 – 28 – 16 remainder = 0 – 6 20 16 so x + 2 is a perfect factor. 3 – 10 – 8 0 decrease the degree by 1 to get the new smaller poly. (3x3 – 4x2 – 28x – 16) / (x + 2) = 3x2 – 10x – 8 – Or – (x + 2) (3x2 – 10x – 8) = 3x3 – 4x2 – 28x – 16