n n – 1 f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0 OFFICIAL POLYNOMIAL JARGON A polynomial function is a function of the form f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0 constant term an  0 an leading coefficient descending order of exponents from left to right. n n – 1 n degree Where an  0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a 0 is the constant term, and n is the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

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

x x2 + 2x - 6 (x + 4) x3 + 6x2 + 2x + 10 + 2x + 10 x3 + 4x2 2x2 2x2 So now, we’re going to use the same process to divide polynomials. determine what to multiply by. Use the 1st term to x x2 + 2x - 6 (x + 4) x3 + 6x2 + 2x + 10 + 2x + 10 x3 + 4x2 2x2 x times what equals x3? 2x2 + 8x - 6x x times what equals 2x2? - 6x - 24 34 x times what equals –6x? This is the remainder! Solution is: x2 + 2x – 6 +

You try . . . 2x2 + 3x + 2 (x + 6) 2x3 + 15x2 + 20x + 10 2x3 + 12x2 x times what equals 2x3? 3x2 + 18x 2x x times what equals 3x2? 2x + 12 -2 x times what equals 2x? This is the remainder! Solution is: 2x2 + 3x + 2 +

Using Synthetic Division 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). What is k if you’re dividing by (x + 7) ? -7 So, k = 3

Using Synthetic Division SOLUTION 2 x 4 + 0 x 3 + (–8 x 2) + 5 x + (–7) Polynomial in standard form Polynomial in standard form 2 0 –8 5 –7 3 k-value 3 • Coefficients Coefficients 6 18 30 105 2 35 6 10 98 Solution! X3-term X2-term Constant Remainder x-term + 2 x3 + 6 x2 + 10 x + 35

You Try . . . Divide 3x4 + 4x3 – 8x – 20 by (x + 2) 3 x 4 + 4 x 3 + 0 x 2 + -8 x + -20 Polynomial in standard form Polynomial in standard form 3 4 0 -8 -20 -2 k-value -2 • Coefficients Coefficients -6 4 -8 32 3 -16 -2 4 12 Solution! X3-term X2-term X-term Constant Remainder - 16 + 3 x3 - 2 x2 + 4 x

--The divisor must be of the form (x – k). 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 x2 + 7x – 8 by (x – 1) 1 7 -8 1 8 Remainder x-term constant x + 8 (x – 1)(x + 8) = x2 + 7x – 8 !! --Something really cool  f(k) = the remainder !!

Try some more. Find f (4) by using synthetic division. f (x) = x2 - 2x - 8 Divide. 3x3 - 2 x2 + 5x - 1 (x + 2) Divide. 4x3 - 7x + 8 (2x - 1)

1 1 1 -14 3 2 • 2 6 14 1 7 3 k-value Solution! Some Vocab Divide. x3 + x2 + x - 14 (x - 2) Polynomial in standard form Polynomial in standard form 1 1 1 -14 3 k-value 2 • Coefficients 2 6 14 1 7 3 Solution! Remainder of zero means (x2 + 3x + 7) is a factor, but is also known as a “Reduced Polynomial”. X2-term Constant Remainder x-term + x2 + 3 x + 7