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# Dividing Polynomials

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1. Dividing Polynomials

2. Simple Division - dividing a polynomial by a monomial

3. Simplify

4. Simplify

5. Long Division - divide a polynomial by a polynomial • Think back to long division from 3rd grade. • How many times does the divisor go into the dividend? Put that number on top. • Multiply that number by the divisor and put the result under the dividend. • Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.

6. x2/x = x -8x/x = -8 x - 8 + 3x -( ) x2 - 8x - 24 -( ) - 8x - 24 0

7. h3/h = h2 4h2/h = 4h 5h/h = 5 h2 + 4h + 5 -( ) - 4h2 h3 - 11h 4h2 -( ) 4h2 - 16h 5h + 28 -( ) 5h - 20 48

8. Synthetic Division - divide a polynomial by a polynomial • To use synthetic division: • There must be a coefficient for every possible power of the variable. • The divisor must have a leading coefficient of 1.

9. Since the numerator does not contain all the powers of x, you must include a 0 for the Step #1: Write the terms of the polynomial so the degrees are in descending order.

10. 5 0 -4 1 6 Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. Since the divisor is x-3, r=3

11. 5 Step #3: Bring down the first coefficient, 5.

12. Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add. 15 5 15

13. 15 45 15 5 Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 41

14. 15 45 123 372 15 41 5 Step #5 cont.: Repeat the same procedure. Where did 123 and 372 come from? 124 378

15. 15 45 123 372 15 41 124 378 5 Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

16. The quotient is: Remember to place the remainder over the divisor.

17. Ex 7: Step#1: Powers are all accounted for and in descending order. Step#2: Identify r in the divisor. Since the divisor is x+4, r=-4 .

18. Step#3: Bring down the 1st coefficient. Step#4: Multiply and add. Step#5: Repeat. 4 -4 20 0 8 -1 1 0 -2 10 -5

19. Ex 8: Notice the leading coefficient of the divisor is 2 not 1. We must divide everything by 2 to change the coefficient to a 1.

20. 3

21. Ex 9: Coefficients 1