1. A polynomial function is a function of the form f (x) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0 Where a n  0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. For this polynomial function, a n is the leading coefficient, a 0 is the constant term, and n is the degree. a n  0 anan anan leading coefficient a 0a 0 a0a0 constant term n n degree descending order of exponents from left to right. n n – 1 OFFICIAL POLYNOMIAL JARGON

2. Let’s remember how to do long division. 7 8 4 0 3 2 6 1 8 6 -1 8 0 4 1 - 3 1 0 3 - 9 1 Remainder = 1 Solution is: 2613

3. So now, we’re going to use the same process to divide polynomials. (x + 4) x 3 + 6x 2 + 2x + 10 x2x2 Use the 1st term to determine what to multiply by. x x times what equals x 3 ? x3x3 2x 2 + 4x 2 x times what equals 2x 2 ? + 2x 2x 2 + 2x + 8x - 6x + 10 x times what equals –6x? - 6 - 6x - 24 34 This is the remainder! Solution is: x 2 + 2x – 6 +

4. You try... (x + 6) 2x 3 + 15x 2 + 20x + 10 2x 2 x times what equals 2x 3 ? 2x 3 3x 2 + 12x 2 x times what equals 3x 2 ? + 3x 3x 2 + 20x + 18x 2x + 10 x times what equals 2x? + 2 2x + 12 -2 This is the remainder! Solution is: 2x 2 + 3x + 2 +

5. Another way to divide a polynomial is to use synthetic division. Use synthetic division to divide 2 x 4 +  8 x 2 + 5 x  7 by (x – 3). This is in the format (x – k). So, k = 3 What is k if you’re dividing by (x + 7) ? -7 Using Synthetic Division

6. Polynomial in standard form Using Synthetic Division 2 x 4 + 0 x 3 + (–8 x 2 ) + 5 x + (–7) 2 6 6 10 18 35 30105 98 20–85 –720–85 –7 Coefficients 3 k -value 3 S OLUTION Polynomial in standard form Remainder Constant + 35 x-term + 10 x X 2 -term + 6 x 2 X 3 -term + 2 x 3

7. Polynomial in standard form 3 x 4 + 4 x 3 + 0 x 2 + -8 x + -20 3 -2 -6 4 4 -16 -832 12 340-8 -20 Coefficients -2 k -value -2 Divide 3x 4 + 4x 3 – 8x – 20 by (x + 2) Polynomial in standard form Remainder Constant - 16 X-term + 4 x X 2 -term - 2 x 2 X 3 -term + 3 x 3 You Try...

8. A Few Notes --The divisor must be of the form (x – k). --If it is not, then you must use long division (i.e. 2x – 7). --If the remainder is 0, then the answer is a factor of the original polynomial! Divide x 2 + 7x – 8 by (x – 1) 1 7 -8 1 1 1 8 8 0 Remainder x-termconstant x + 8 (x – 1)(x + 8) = x 2 + 7x – 8 !! --Something really cool  f(k) = the remainder !!

9. Try some more. Divide. x 2 - 2x - 8 (x - 4) Divide. 3x 3 - 2 x 2 + 5x - 1 (x + 2) Divide. 4x 3 - 7x + 8 (2x - 1)