Division of Polynomials

Transcription

1 Division of Polnomials Dividing Monomials: To divide monomials we must draw upon our knowledge of fractions as well as eponent rules. 1 Eample: Divide. Solution: It will help to separate the coefficients and variables as follows. 1 1 Now, divide the coefficients and use the quotient rule of eponents on the variables z Eample: Divide 1 z. Solution: It will help to separate the coefficients and variables as follows. 18 z 18 z 1 z 1 z Now, divide the coefficients and use the quotient rule of eponents on the variables. 18 z 1 z Sometimes division problems are actuall written with the division sign. If so, first rewrite in fraction form and proceed as with the previous problems. z 0 Eample: Find the quotient: 0 p q pq Solution: Write the problem in fraction form, and then simplif using eponent rules. 0 p 0 p q pq p q q pq

2 Eample: Find the quotient: 7 1 Solution: Write the problem in fraction form, and then simplif using eponent rules Dividing a Polnomial b a Monomial: This is simpl an etension of division of a monomial b a monomial. Use the distributive propert to divide the monomial (denominator) into each term of the polnomial (numerator) Eample: Simplif the quotient: 7 Solution: Divide each term of the polnomial in the numerator b the monomial in the denominator Eample: Simplif the quotient: Solution: Divide each term of the polnomial in the numerator b the monomial in the denominator

3 10 p q 1 pq 9 p q Eample: Divide: p q 7 Solution: Divide each term of the polnomial in the numerator b the monomial in the denominator p q 1 pq 9 p q p q 10 p q p q p 1 pq p q q pq 9 p q p q 7 Eample: Simplif the quotient: (a bc 1ab c 0abc) (10a b c) Solution: First, write the problem in fraction form. (a bc 1ab 0abc) (10a b c) a bc 1ab c 0abc 10a b c 10a b c 10a b c c c ab a a b c Dividing a Polnomial b a Binomial: The procedure for dividing a polnomial b a binomial is similar to that of dividing whole numbers, a process commonl called long division. Before we get started, it is necessar to review some basic terminolog. 1. Divisor The number being divided into another number. Dividend The number that is being divided into.. Quotient The resulting answer. 1 Eample: In the division problem, identif the divisor, dividend, and quotient. Solution: In this eample is the divisor, is the dividend, and 1 is the quotient. Eample: In the division problem 7 1, identif the divisor, dividend, and quotient. Solution: In this eample is the divisor, 7 1 is the dividend, and is the quotient.

4 Inverse Nature of Division and Multiplication: Remember that division and multiplication are inverse operations, that is, the undo each other. Consequentl, ou ma use multiplication to check our answer to a division problem b multipling the divisor b the quotient to obtain the dividend as demonstrated in the net eample. Eample: Check the division problem (Divisor)(Quotient)Dividend 1 b using multiplication. Solution: Multipl the divisor b the quotient. If the product equals the dividend, then the answer checks. (Divisor)(Quotient)Dividend 1 The answer checks. Eample: Check the division problem 7 1 b using multiplication. Solution: Multipl the divisor b the quotient. If the product equals the dividend, then the answer checks. (Divisor)(Quotient)Dividend ( )( ) 7 1 Once again, the answer checks Long Division of Polnomials: As previousl stated, the procedure for dividing polnomials is ver similar to the procedure for dividing whole numbers. I will outline the procedure step b step in the following eample. Eample: Divide 7 1 b. Solution: Before getting started, set up the problem as a long division problem. Be careful when identifing the divisor and dividend. 7 1 Step 1: Divide the 1 st term in the dividend b the 1 st term in the divisor [ ]. Write the in the quotient. 7 1 Step : Multipl the in the quotient b the divisor [ ( ) ]. Write the product on the second line under the dividend.

5 7 1 Step : Subtract ( ) from ( 7 1). Since we are not subtracting anthing from the 1, we simpl bring it down. The last line is called the remainder. Since the remainder is a 1 st degree polnomial and the divisor is a 1 st degree polnomial we are not done Repeat these steps until the remainder is either 0 or one degree less than the divisor. Repeat Step 1: Divide the 1 st term in the remainder b the 1 st term in the divisor ( ). Write the - in the quotient Repeat Step : Multipl the - in the quotient b the divisor ( ( ) ). Write the product on the line under the remainder Repeat Step : Subtract ( 1) from ( 1). Since the remainder is 0, the divisor divide into the dividend evenl and there is no remainder. Alwas check our answer b multipling the quotient b the divisor. This was alread done in a previous eample. 7 1 Eample: Divide b Solution: I will complete this eample step b step, but without eplanation. Step 1: Divide

6 Step : Multipl Step : Subtract Repeat Step 1: Divide Repeat Step : Multipl Repeat Step : Subtract Verif that the answer checks ( )( 1) Descending Order: When performing long division of polnomials the divisor and the dividend must be written in descending order. Eample: Divide 1s 10 9s b s. Solution: Since neither the dividend nor the divisor is written in descending order, the must both be rewritten when setting up the long division.

7 s 9s 1s 10 I will now complete the problem but without showing each step. Instead, I am doing the steps mentall. Be sure ou understand each step. s s 9s 1s 10 9s s... 1s s Verif that the answer checks. (s )(s ) 9s 9s 1s 10 9s 1s 10 1s 10 Place Holders for Missing Terms: In long division of polnomials, ever power of the variable must have a place in the dividend from the highest power of the variable on down. If this term is missing, we put in a place holder. Eample: Divide 19 b. Solution: Notice that in the dividend, there is no term with an to the first power. Since this term is missing, we must put in a place holder. In this case, the place holder is 0. Long division is then set up as follows: 19 0 Again, I will do most of the work mentall. Make sure ou understand what is happening at each point in the problem There is a remainder of (-1). To check a solution with a remainder we need to add the remainder to the product of the quotient and divisor. A simple formula to remember is: (Quotient)(Divisor)RemainderDividend

8 ) ( ) )( ( Eample: Find the Quotient: ) ( ) (7 Solution: Here is the whole problem Dividing a Polnomial b a Polnomial: So far all the long division eamples have used a binomial in the divisor. We can use long division to divide a polnomial b an other polnomial. Eample: Divide b 1 Solution: This eample will involve the same steps of long division onl the will be repeated more times Eample: Find the quotient: ) ( ) ( Solution:

9 The remainder is Check using the formula: (Quotient)(Divisor)RemainderDividend ( )( ) ( ) Applications: Eample: A prepaid phone card compan charges a $.0 connection fee for each call plus $0.10 for each minute. Find the average cost per minute of each call in terms of, where is the number of minutes per call. What is the average cost per minute of a call that lasts 10 minutes? Solution: The cost of a call is the total of the connection fee plus the cost per minute times the number of minutes which is represented as The average cost per minute ma be determined b dividing b the number of minutes Simplif this b dividing The average cost per minute for a 10 minute call is: The average cost per minute for a 10 minute call is $0..

10 Eample: A homeowner wishes to increase the length of his patio b twice as much as the width. The area of the new patio is represented b the polnomial epression 0 8. If the original length was 8 feet, find the width of the new patio in terms of. If the homeowner increases the width of the patio b 1 foot, what are the dimensions of the new patio? Solution: Let the amount of increase in the width of the patio. Then the amount of increase in the length. The new length can then be represented b 8. The formula for the area of a rectangle is A LW. Likewise, we can solve this for width to obtain W A L We can now solve for W b substituting values for A and L. 0 8 W 8 This requires long division The quotient is, therefore the width of the patio is represented b W. If the homeowner increases the width of the patio b 1 foot then the dimensions are as follows: W 1 7 L 8 (1) 8 10 The dimensions of the new patio are 7 b 10.