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Name:_________________ Date:________ Polynomial Division We spent yesterday talking about a way to factor polynomials that had been created by taking a binomial to a power: ( ax+ b) n , but what about polynomials that are created by multiplying different binomials together? We need to come up with some way to factor these more complex polynomials. Let’s take a look at how we find factors with numbers and then maybe we can figure out a good way to find factors of polynomials. Exercise 1: Your little brother is learning about long division from Yumi- sensei. A problem on his homework looks like this: Find the prime factorization of 576. You want to teach him how to do this, not just tell him the answer. (a) What is the very first thing you think he should do to try to solve this problem? Why do you thing this thing should be done first? Which numbers would you choose? (b) How can you tell if a factor you guessed is really a factor of 567? (c) Go ahead and solve the problem yourself. (d) What tool (you had to use it several times) helped you find all the prime factors of 576? (e) Think about the tool that you used above. Can you come up with a list of steps showing how to use this tool? A list of steps like this would help your brother so that he doesn’t forget to subtract, or bring down the next number.

Polynomial Division Exploratory Ws

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This is a worksheet designed to help students discover polynomial division by comparing it to regular long division

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Page 1: Polynomial Division Exploratory Ws

Name:_________________Date:________

Polynomial DivisionWe spent yesterday talking about a way to factor polynomials that had been created by taking

a binomial to a power: (ax+b )n, but what about polynomials that are created by multiplying

different binomials together? We need to come up with some way to factor these more complex polynomials. Let’s take a look at how we find factors with numbers and then maybe we can figure out a good way to find factors of polynomials.

Exercise 1: Your little brother is learning about long division from Yumi-sensei. A problem on his homework looks like this: Find the prime factorization of 576. You want to teach him how to do this, not just tell him the answer.

(a) What is the very first thing you think he should do to try to solve this problem? Why do you thing this thing should be done first? Which numbers would you choose?

(b) How can you tell if a factor you guessed is really a factor of 567?

(c) Go ahead and solve the problem yourself.

(d) What tool (you had to use it several times) helped you find all the prime factors of 576?

(e) Think about the tool that you used above. Can you come up with a list of steps showing how to use this tool? A list of steps like this would help your brother so that he doesn’t forget to subtract, or bring down the next number.

(f) So you and your brother have been doing the problem separately and he gets 2732. Rather than telling him he’s right or he’s wrong, you want to encourage him to check his answers for himself and not rely on someone else. How can he check his answer?

Page 2: Polynomial Division Exploratory Ws

Exercise 2: Now you turn to your own math homework after helping your little brother. The problem asks: find all the factors of the polynomial x3+4 x2+x−6. Wait a second! You’ve never learned how to do this. You start hyperventilating and break into a cold sweat. DeLacy-sensei is going to kill you. You reach back and think. You learned how to factor things with x2 in them, but it doesn’t seem like that’s going to work in this case. You start feeling pretty ill. Well, wait a second, you know DeLacy-sensei wouldn’t give you a problem you can’t solve. You know that to produce a polynomial like this, you’d have to multiply three binomials together, so you decide to multiply 3 random binomials together to see what you get, and to see if you notice any patterns.

(a) Try making up three binomials. They should all be in the form (x ±a). Multiply them together.

(b) Look at the original three binomials and the polynomial they produced when they were multiplied together. Do you see anything interesting- like perhaps, the very last number that’s by itself. Which part of the binomials created that last number?

(c) Use this discovery to try to factor x3+4 x2+x−6. Trial and error are your friends.

Exercise 3: Whew. You did it. Yay! You’re not going to die after all. Now you look at the next problem on your homework: Factor x3−12 x2+20 x+96. You start to see black spots you’re hyperventilating so much. NINETY-SIX!? There’s no way you can try all the different things that multiply to be 96. We have ±1 ,±2 ,±3 , ±4 , ±6 , ±8 ,±12 ,±16 , ±24 , ±32 , ±48∧±96. Trial and error are very much NOT your friends anymore. You will never go out for pizza with them again. Before you pass out though, you think back to what you were doing with your little brother earlier in the evening. He had a really big number to factor, and he did it using a very powerful tool called long division. Maybe, just maybe, there’s a way to apply long division in this instance.

(a) Choose three binomials that you think are most likely factors of this polynomial. Why do you think these factors could be factors of the binomial x3−12 x2+20 x+96?

Page 3: Polynomial Division Exploratory Ws

(b) Set up a division problem just like we did for 576. Put the big polynomial on the inside, and put the thing you think is a factor on the outside.

(c) Now you’re stuck, you’re not sure what to do next. Well, notice that we’re dividing something with two parts into something with four parts. Take a look at the following division with numbers: 52¿4650. Notice how we can’t do 52 into 4, and we can’t do 52 into 46, but we can do 52 into 465. Now, when we try to guess how many times 52 goes into 465, it would be much easier to think of 52 as about 50 and think about how many times 50 goes into 450- 9 times. Concentrating just on the 5, not on the 2 makes estimating this division much easier. Try this same logic with our polynomial division problem. You should have something that looks like this: x±a¿ x3−12 x2+20x+96. Let’s

disregard the thing added to x and concentrate just on the xs. How many times does x go into x3? Or in other words, what do you need to multiply x by to get x3?

(d) Now, where should you put this result? In the division problem 52¿4650, we know that 52 goes into 465 nine times and we would put the 9 above the 5, not above the 4 or the 6. So thinking about the fact that x goes into x to the third x2 times, where should we

put this x2? Where is the x2’s place if we’re thinking in terms of place value?

(e) Now, it seems like the rules for regular division should now apply. Go back and take a look at the list of steps you created for regular division and see if they’ll work with this new polynomial division you’ve just created. What do you think your next three steps in your polynomial division problem should be?

(f) Go ahead and try to see if the factor you chose really is a factor of the polynomial x3−12 x2+20 x+96. What should you get at the end of your division that will confirm that the factor you chose is, indeed a factor?

(g) Here’s where it gets fun. If you didn’t get a remainder of zero that means your initial guess was incorrect. You need to keep trying factors until one gives you a remainder of zero. So try the other two factors you thought would be good possibilities. If those don’t work, try others. There’s plenty of space below.

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(h) Now that you’ve found one factor, what should you do to find the others? Go back and look at the problem you did with your little brother.

(i) Give a complete list of all the factors of x3−12 x2+20 x+96 below.

Exercise 4: Let’s practice this new polynomial division skill you created. Please keep in mind that the steps are the same as for regular division.

(a) Find all the factors of x3+2x2−31 x+28

(b) Find all the factors of 2 x3−3 x2−8 x−3 (Don’t let that 2 confuse you, just do it the same way you did before.)

(c) Find all the factors of x3−19 x−30 (Note the fact that there’s no x2 term, but in terms of

place value, you need there to be a spot for x2 terms to go. What should you do?)

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You can rest easy now knowing that DeLacy-sensei won’t kill you tomorrow.