Hw - Polynomial Long Division

Dividing Polynomials 1. The problem

Name: __________________

28k p − 42kp + 56kp does not require polynomial long division. 14kp 3

2

3

a. Why not?

b. Write down the answer. This should require little-to-no work.

2. Suppose you wanted (for some reason) to factor the polynomial f(x)=2x3+x2−13x+6. It’s not obvious how to factor it—or even whether it can be factored—is it? But watch. a. Find f(2). (This shouldn’t take too long, even without a calculator.) b. Based on your work in Part (a), you now know that one of the factors of f(x) is one of the following. Which one? (Check one.)  (x-2)  (x+2)  (2x) 𝑥

 (2 ) c. Since 2 is a zero of the function, the answer to part b should be a factor, so what remainder are you expecting to get when you divide? 2𝑥 3 +𝑥 2 −13𝑥+6

d. Now, actually divide using polynomial long division: 𝑡ℎ𝑒 𝑎𝑛𝑠𝑤𝑒𝑟 𝑦𝑜𝑢 𝑐ℎ𝑒𝑐𝑘𝑒𝑑 𝑖𝑛 𝑃𝑎𝑟𝑡 (𝑏). Your answer is another polynomial.

e. Factor the answer that you got to Part (c).

f. Now you’re ready: write down the complete factored form of 2x3 + x2 − 13x + 6.

Simplify using polynomial long division. Write remainders, if they exist, as a fraction added to the quotient. 3.

x 2 − 12 x − 45 = x+3

4. a.

2 y 2 + y − 16 = y −3

b. Choose a number for y. Plug it into my original function, and also into your simplified expression. Make sure you get the same answer! (You can test all your answers this way, but in this case I want you to actually write down what you tested and what you got.)

2h 3 − 5h 2 + 22h + 51 5. = 2h + 3

6. a.

2 x 3 − 4 x 2 + 6 x − 15 = x2 + 3

b. Demonstrate that your answer is correct by multiplying back.

7. In Problem 4 you divided 2y2+y-16 by y-3. a. Plug the number y=3 into the function 2y2+y-16.

b. The number you got in Part (a) also showed up in your solution to Problem 4. Where?

8. This problem is brought to you by the fraction

3𝑥 3 +4𝑥 2 −15𝑥+11 𝑥−2

.

a. First, do the polynomial long division.

b. Now, plug x=2 into the function 3x3+4x2-15x+11.

c. The number you got in Part (b) also showed up in your solution to Part (a). Where?

9. Problems 7 and 8 were meant to demonstrate an important and very general rule. State that rule. (This will involve some words and some math symbols.)