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Dividing Polynomials, Part 1 Section 5.6A Dividing Polynomials, Part 1

Section 5.6 Part 1 Dividing a polynomial by a monomial: Example Divide each term of the polynomial separately by the monomial. Example

Problem from today’s homework: Divide. Simplify your answer. Do not factor. 12 𝑥 2 𝑦 2 +4𝑥 𝑦 2 −16 𝑦 2 4 𝑥 2 𝑦 . 𝟑𝒚+ 𝒚 𝒙 − 𝟒𝒚 𝒙 𝟐

Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers. This process is reviewed in detail on the next slide, but first, try these two simpler examples in your notebook (use long division, not your calculator): 1). 225 ÷ 9 (Answer: 25) 2). 232 ÷ 9 (Answer: 25 with a .. remainder of 7, or 25 + 7/9) Question: How can you check your answers on long division problems? (A: Multiply answer times divisor, e.g. 25 * 9 = 225)

Example: Long Division with integers Divide 43 into 72. Multiply 1 times 43. Subtract 43 from 72. Bring down 5. Divide 43 into 295. Multiply 6 times 43. Subtract 258 from 295. Bring down 6. Divide 43 into 376. Multiply 8 times 43. Subtract 344 from 376. Nothing to bring down. 32 is smaller than 43, so we are done. 32 43 We then write our result as 168

As you can see from the previous example, there is a pattern in the long division technique. Divide Multiply Subtract Bring down Then repeat these steps until you can’t bring down or divide any longer. We will incorporate this same repeated technique with dividing polynomials.

Now you try it (And don’t forget to check your answer!) Divide 3473 by 6 using long division. Then check your answer. Do this in your notebook now, and make sure you ask if you have questions about any step. This will be crucial to your understanding of long division of polynomials. Answer: 578 5 6 (Can also be written as 578 + ) 5 6

15 - 35 - x Example with polynomials: So our answer is 4x – 5. Divide 7x into 28x2. Multiply 4x times 7x+3. Subtract 28x2 + 12x from 28x2 – 23x. Bring down -15. 15 - Divide 7x into –35x. 35 - x Multiply -5 times 7x+3. Subtract –35x–15 from –35x–15. Nothing to bring down. So our answer is 4x – 5. Check: Multiply (7x + 3)(4x – 5) and see if you get 28x2 – 23x - 15.

Now you try it (And don’t forget to check your answer!) Divide 6x2 – x – 2 by 3x – 2 using long division. Then check your answer. Do this in your notebook now. ANSWER: 2x + 1 Check: Multiply (2x + 1)(3x – 2). What do you get?

10 - 8 6 4 7 2 + - x x 14 4 + 8 + 20 - x 70 20 - x 78 10 - x 2 Example Divide 2x into 4x2. 8 6 4 7 2 + - x Multiply 2x times 2x+7. Subtract 4x2 + 14x from 4x2 – 6x. x 14 4 2 + 8 + Bring down 8. 20 - x Divide 2x into –20x. 70 20 - x Multiply -10 times 2x+7. Subtract –20x–70 from –20x+8. 78 Nothing to bring down. + ) 7 2 ( 78 x 10 - We write our final answer as

How do we check this answer? x 2 10 - 8 6 4 7 2 + - x Final answer: 2x – 10 + 78 . 2x + 7 x 14 4 2 + 8 + 20 - x 70 20 - x 78 How to check: Calculate (2x + 7)(2x – 10) + 78. If it comes out to 4x2 – 6x + 8, then the answer is correct.

Now you try it (And don’t forget to check your answer!) Divide 15x2 + 19x – 2 by 3x + 5 using long division. Then check your answer. Do this in your notebook now. Answer: 5x – 2 + 8 . 3x + 5

REMINDERS: Homework Questions? Use the Open Lab! The assignment on today’s material (HW 5.6A) is due at the start of the next class session. Homework Questions? Use the Open Lab!

and begin working on the homework assignment. You may now OPEN your LAPTOPS and begin working on the homework assignment. We expect all students to stay in the classroom to work on your homework till the end of the 55-minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.