3.2 Polynomials-- Properties of Division Leading to Synthetic Division

Part 1 Some serious relationships

Start with a POD Using long division, divide x by x 2 + 3x + 1 By hand. On CAS.

Start with a POD Using long division, divide x by x 2 + 3x + 1 Quotient: x 2 - 3x + 8 Remainder: -21x - 24

The POD leads to The Division Algorithm for Polynomials: f(x) = d(x) · q(x) + r(x) where d, q, and r are polynomials, d is the divisor, q is the quotient, r is the remainder, and r(x) has a smaller degree than d(x).

There’s a pattern– what is it? Find the remainders with LINEAR divisors: Compare that to f(3)– the top polynomial. Compare that to f(-4).

The Relationship The Division Algorithm for Polynomials: f(x) = d(x) · q(x) + r(x) What happens when the remainder = 0? The Factor Theorem: If the remainder = 0, then f(c) = 0. If f(c) = 0, then c is a root/ solution/ zero of f(x) and (x - c) is a factor of f(x).

The Proof Start with: f(x) = d(x) · q(x) + r(x) What if d(x) = x - c ? (This is a linear divisor.) f(x) = (x - c)q(x) + r(x) r(x) must be of a degree less than (x - c), so r(x) must be a constant, k Let’s see what happens when we plug c in for x to find f(c): f(c) = (c - c)q(c) + k = 0·q(c) + k In other words, f(c) is the remainder k. The Remainder Theorem: If polynomial f(x) is divided by x - c, the constant remainder r(x) is f(c).

Use it 1. If f(x) = x 3 - 3x 2 + x + 5, use the Remainder Theorem to find f(-4). 2.Show that (x - 2) is a factor of f(x) = x 3 - 4x 2 + 3x Find a polynomial f(x) that has zeros at x = 2, -1, and 3. (How many could you find? How many that are 3 rd degree?

Use it 1. If f(x) = x 3 - 3x 2 + x + 5, use the Remainder Theorem to find f(-4).Divide f(x) by (x+4). 2.Show that (x - 2) is a factor of f(x) = x 3 - 4x 2 + 3x + 2. Show that f(2) = 0. 3.Find a polynomial f(x) that has zeros at x = 2, -1, and 3. f(x) = (x-2)(x+1)(x-3) (How many could you find? How many that are 3 rd degree? The answer to both of these is “infinite.” Consider f(x) = (x- 2)(x+1)(x-3) 2 to illustrate the first answer, and f(x) = 3(x- 2)(x+1)(x-3) to illustrate the second answer.)

Part 2 Let’s make it easier.

Dividing polynomials Use long division (not the easy part):

Dividing polynomials Use long division: With a remainder of 25. What is f(-3)?

Dividing polynomials Now, use synthetic division: becomes -3 | |____________ What is the remainder? What is the quotient?

Use this to find zeros of polynomials What polynomials are represented below? What is the remainder? The quotient? What is the full factorization over the set of integers? What does it all mean? -11| |____________

Use this to find zeros of polynomials What polynomials are represented below? What is the remainder? The quotient? 0 x 2 - 3x + 4 What does it all mean? For f(x) = x 3 + 8x x is a zero/ solution/ root (x + 11) is a factor of f(x) What are the integer factors of f(x)? (x + 11)(x 2 - 3x + 4)

Use this to find other information about polynomials For f(x) = x 3 + 8x x +44 use synthetic division to find f(2).

Use this to find other information about polynomials For f(x) = 2x 4 + 5x 3 - 3k, What is k if f(2) = 39?

Summary (using different letters) If f(a) = b, then 1. b is the remainder of f(x)/(x - a). 2. (a, b) is on the graph of f(x). 3. The value of f(x) at a is b. If b = 0 also, then 1. (x - a) is a factor of f(x). Note: linear factor! 2. a is a zero of f(x). 3. (a, 0) is on the graph of f(x) 4. a is an intercept of f(x). 5. a is a solution of f(x) = 0.

Use it one more time 1.Give a factor of f(x) = x 3 - 5x 2 - 6x if (6, 0) is on the graph. What is another factor? 2.If (-4, k) is on the graph of x 3 - 2x + 2, what is k? How many ways could you corroborate your answer?