Expert: Richard J. Raridon - 7/5/2011

Question
Divide:

x2 - x + 3
x + 1  
x + 2 +  5  
x + 1
 x - 2 +  5  
x + 1

x + 2  x + 2 +  5  
x - 1


2. Divide:

9z - z2
z - 3
-z + 6 +  18  
z - 3
 z + 6 +  18  
z - 3

z + 24  z - 24

3. Divide:  

6u2 + 7u + 5
3u - 1  
2u + 3 +  8  
3u - 1
 3u + 2 +  8  
3u - 1

2u = 11  3u - 6i

4. Divide:

z3 + 3z2 - 13z + 6
 z - 2   
z + 5  z2 + 5z
z2 + 5z - 3  z2 - 3

5. Divide using synthetic division:

2x3 - 4x2 - 7x + 5
 x - 3   
2x2 + 2x - 1  2x2 + 2x - 1 +  2  
x - 3

2x2 + 2x +  2  
x - 3
 2x2 +  2  
x - 3


6. Divide using synthetic division:

3x3 - 2x2 + x + 4
 x + 1   
3x2 - 5x + 4  3x2 - 5x -  2  
x + 1

3x2 + 5x + 4  3x2 - 5x + 6 +  -2  
x + 1


7. Divide using synthetic division:

2u4 - 5u3 - 12u2 + 2u - 8
  u - 4    
2u3 + 3u2 + 2  2u4 + 3u2 + 2u
2u2 + 3u + 2  2u3 + 3u2 - 2

8. Divide using synthetic division:

y4 - 4y2 + y + 4
 y + 2   
y2 - 2y +  2  
y + 2
 y3 - 2y2 + 3
y3 + 2y2 + 3  y3 - 2y2 + 1 +  2  
y + 2


9. Divide using synthetic division:

x6 - 1
x + 1
x4 + x3 - x2 + x - 1  x5 - x4 + x3 - x2 + x - 1
x5 - x4 - x3 + x2 - 1  x3 - x2 - 1

10. Use synthetic substitution to find P(c) for the given polynomial P(x) and the given number x:

P(x) = 4x3 - 2x2 + x - 1; c = -0.5

P(-0.5) = 3.5  P(-0.5) = 2.5
P(-0.5) = -3.5  P(-0.5) = -2.5

11. Use synthetic substitution to find P(c) for the given polynomial P(x) and the given number x:

P(x) = x3 + 4x2 - 8x - 6; c = -5

P(-5) = 9  P(-5) = -12
P(-5) = 759  P(-5) = 6

12. Use synthetic substitution to find P(c) for the given polynomial P(x) and the given number x:

P(x) = 1 - 7x - 4x2 + x3; c = 6

P(6) = 115  P(6) = 114
P(6) = 31  P(6) = 42

13. Use synthetic substitution to find P(c) for the given polynomial P(x) and the given number x:


P(x) = 6x3 - x2 + 4x + 3;    c = - 1
3
P(- 1 ) =  16
3 3
 P(- 1 ) =  5
3 2

P(- 1 ) =  4
3 3
 P(- 1 ) =  9
3 8


14. Find the remainder when t + 1 is divided into

P(t) = t5 + t4 + t3 + t2 + t + 1

-2  1
0  3

15. Find the remainder when z + 2 is divided into

P(z) = z5 + 2z4 + z3 + 2z2 + z + 2

2  1
3  0

16. Solve using the given root:

2x3 + 9x2 + 7x - 6 = 0; -2

{-2,0.5, 3}  {-2,0.5,-3}
{-2,-0.5,-3}  {-2,1,0}

17. Solve using the given root:

2z3 + z2 - 8z + 3 = 0; 1.5

-1 ± Ö 2
 -1 ± 2 Ö 2

-3 or 1  3 or -1

18. Find a polynomial equation with integral coefficients that has the roots:

-2, 2, -3

x3 + x2 - 4x - 12 = 0  x3 + 3x2 - 4x - 12 = 0
x3 + x2 - 8x - 12 = 0  x3 - 3x2 + 4x - 12 = 0

19. Find a polynomial equation with integral coefficients that has the roots:

-2, -i, i

x3 + x + 2 = 0  x3 + 2x2 + 2 = 0
x3 + 2x2 + x + 2 = 0  x3 - 2x2 - x - 2 = 0

20. Find a polynomial equation with integral coefficients that has the roots:

1, -1.5, 2

2x3 - 3x2 - 5x + 6 = 0  2x3 + 3x2 - 7x + 6 = 0
2x3 - x2 - 5x + 6 = 0  2x3 + 3x2 - 5x - 6 = 0

21. Solve given the two roots:

2x4 - 5x3 - 11x2 + 20x + 12; -2, 3

{-0.5, 2}  {0.5, 2}
{±2, 3, -0.5}  {0, 3}

22. Find an equation with integral coefficients that has the given roots:

1, -3, -4

x3 + 12x2 + 5x - 12 = 0  x3 + 6x2 + 5x - 12 = 0
x3 - 6x2 + 5x - 12 = 0  x3 - 12x2 + 10x - 24 = 0

23. Determine the missing roots:

Equation:  x4 - 3x3 + 4x2 - 6x + 4 = 0

Roots:   

1, 2, iÖ 2
-iÖ 2
 -iÖ 5

-i  iÖ 2


24. Solve given the root indicated:

Equation:  x4 - 5x2 - 10x - 6 = 0

Root:  -1 + i

{3, 1, -1, ±i}  {-1 ± i, 3, -1}
{3, -1, -1 + i}  {-1, -i, i, 1}

25. Solve for x:

x + 1 =     2
-x + 2  x
x = ±4  x = 1, 4
x = 1, -4  x = -1, 3

26. Solve:

|a/2+1| = 3

a = -2 or 1  a = -2 or 4
a = -4 or 8  a = -8 or 4

27. Solve for w:

w2 + 7w = 1
2 4
w = 0.5 or -4  w = 0 or -4
w = -0.5 or 4  w = 0 or 4

28. Solve for n:  

Ö 5n + 6 = n
n = -1  n = 0
n = -3  n = 6

29. Solve for n:

Ö 2n + 4 = 6
n = 12  n = -2
n = -16  n = 16

30. Solve for n:

Ö 4n - 8 = 10
n = 21  n = 20
n = 22  n = 27

Part 2
Type the answer to the problem in the text box below each item. Be sure legibly show all work in your notebook. Remember to include any applicable units. If there is no solution, type "no solution". If there is not enough information present to solve the problem, type "not enough information". (Each question is worth two points)
31. Divide:

4u4 - 4u3 - 5u2 - 9u - 1
 2u - 1       


32. Divide:

2a3 + 8a2 - 3a + 8
a + 4



33. Divide:

2x4 - x3 + x2 + 4x - 4
2x2 + x - 2


34. Find f(c) for the given polynomial f(x) and value c:

f(x) = 3x3 + 8x2 - 6x + 4; c = -4


35. Find the remaining roots for the following equation:


2x3 - x2 + 10x - 5 = 0; iÖ 5

Answer
It took me a little while to realize some of those numbers meant remainders
1. x-2, remainder 5
2. -z+6, rem. 18
3. 2u+3, rem. 8
4. z^2+5z-3
5. 2x^2+2x+1, rem. 2
6. 3x^2-5x+6, rem. -2
7. 2u^3+3u^2+2
8. y^3-2y^2+1, rem. 2
9. x^5-x^4+x^3-x^2+x-1
10. -2.5
11. 9
12. 31
13. p(-1) = -8
14. 0
15. 0
16. {-2,0.5,-3}
17. -1 +/-(2^1/2)
18. x^3+3x^2-4x-12=0
19. x^3+2x^2+x+2=0
20. 2x^3-3x^2-5x+6=0
21. {+/-2,3,-0.5}
22. x^3+6x^2+5x-12=0
23. -i(2^1/2)
24. {-1+/-i,3,-1}
31. no solution
32. no solution
33. x^2-x-2
34. f(-4) = -36
35. x = 0.5,-i(5^1/2)