Lesson 6: Dividing by X-A and by X+A

M1 CC A2 L6

Lesson 6: Dividing by x-a and by x+a

Homework

Introductory Exercise

Find the following quotients, and write the quotient in standard form. Be prepared to present solutions to the class.

a.  x2-9x-3 b. x3-27x-3

c.  x4-81x-3

What patterns do you notice in the Opening Exercise?

Use the patterns you observed in the Opening Exercise to determine the quotient of x5-243x-3. Explain your reasoning. Test your conjecture using one of the division methods we have discussed in class.

Exercise 1

1.  Use patterns to predict each quotient. Explain how you arrived at your prediction, and then test it by applying the reverse tabular method, long division, or synthetic division.

a.  x2-144x-12 b. x3-8x-2

c.  x3-125x-5 d. x6-1x-1

Example 1

What is the quotient of x2-a2x-a? Use the reverse tabular method or long division.

Exercises 2–4

2.  Work with your group to find the following quotients.

a.  x3-a3x-a b. x4-a4x-a

What patterns do you notice in the quotient?

How do these patterns compare to the ones observed in the opening exercise?

3.  Predict without performing division whether or not the divisor will divide into the dividend without a remainder for the following problems. If so, find the quotient. Then check your answer.

a.  x2-a2x+a b. x3-a3x+a

c.  x2+a2x+a d. x3+a3x+a

4. 

a.  Find the quotient xn-1x-1 for n=2, 3, 4, and 8

b.  What patterns do you notice?

c.  Use your work in part (a) to write an expression equivalent to xn-1x-1 for any integer n>1.

Problem Set

1.  Compute each quotient.

a.  x2-625x-25 / b.  x3+1x+1
c.  x3-18x-12 / d.  x2-0.01x-0.1

2.  In the next exercises, you can use the same identities you applied in the previous problem. Fill in the blanks in the problems below to help you get started. Check your work by using the reverse tabular method or long division to make sure you are applying the identities correctly.

a.  16x2-1214x-11=___2-___24x-11=___+11

b.  25x2-495x+7=___2-___25x+7=___-___= ______

c.  8x3-272x-3=___3-___32x-3=___2+______+___2=______

3.  Show how the patterns and relationships learned in this lesson could be applied to solve the following arithmetic problems by filling in the blanks.

a.  625-8116=___2-9225-___=___+___=34

b.  1000-277=___3-___3___-3=___2+10___+___2=______

c.  100-97=___2-___2___-3=______

d.  1000+6414=___3+___3___+___=___2-______+___2=______

4.  Apply the identities from this lesson to compute each quotient. Check your work using the reverse tabular method or long division.

a.  16x2-94x+3

b.  81x2-2518x-10

c.  27x3-83x-2

5.  Extend the patterns and relationships you learned in this lesson to compute the following quotients. Explain your reasoning, and then check your answer by using long division or the tabular method.

a.  8+x32+x / b.  x4-y4x-y
c.  27x3+8y33x+2y / d.  x7-y7x-y