SOLUTION: If the equations x^3-(a+2)x+2b=0 and 2x^3+ax^2-4x-b=0 have a common factor (x+3), find the values of a and b and find a second common factor.

Question 366517: If the equations x^3-(a+2)x+2b=0 and 2x^3+ax^2-4x-b=0 have a common factor (x+3), find the values of a and b and find a second common factor.
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
If the equations x^3-(a+2)x+2b=0 and 2x^3+ax^2-4x-b=0 have a common factor (x+3), find the values of a and b and find a second common factor.

..........x^2 - 3x - (a + 2) + 9
x + 3 --> x^3 + 0x^2 - (a + 2)x + 2b
..........x^3 + 3x^2
..............- 3x^2 - (a + 2)x
..............- 3x^2 - 9x
.....................- (a + 2)x + 9x + 2b
.....................- (a + 2)x - (a + 2)3
................................9x + 2b + (a + 2)3
................................9x + 27
.....................................2b + (a + 2)3 - 27

..........2x^2 + ax - 6x + 14 - 3a
x + 3 --> 2x^3 + ax^2 - 4x - b
..........2x^3 + 6x^2
...............ax^2 - 6x^2 - 4x
...............ax^2 + 3ax
....................-6x^2 - 3ax - 4x
....................-6x^2 - 18x
.........................14x - 3ax - b
.........................14x + 42
.............................-3ax - b - 42
.............................-3ax - 9a
.................................-b + 9a - 42

2b + (a + 2)3 - 27 = 0
-b + 9a - 42 = 0

2b + 3a + 6 - 27 = 0
3a + 2b - 21 = 0
3a + 2b = 21
3a = 21 - 2b
a = 7 - (2/3)b
2b = 21 - 3a
b = (21/2) - (3/2)a

-b + 9a - 42 = 0
9a = b + 42
a = (1/9)b + (42/9)
-b = 42 - 9a
b = 9a - 42

set a's equal and solve for b
7 - (2/3)b = (1/9)b + (42/9)
- (2/3)b - (1/9)b = (42/9) - 7
- (6/9)b - (1/9)b = (42/9) - (63/9)
- (7/9)b = - (21/9)
(7/9)b = 21/9
7b = 21
b = 3

plug b = 3 into a = 7 - (2/3)b
a = 7 - (2/3)3
a = 7 - 2
a = 5
plug b = 3 into a = (1/9)b + (42/9)
a = (1/9)3 + 42/9
a = 3/9 + 42/9
a = 45/9
a = 5

a = 5, b = 3

plug a into these 2:
x^2 - 3x - (a + 2) + 9
2x^2 + ax - 6x + 14 - 3a

x^2 - 3x - (5 + 2) + 9
x^2 - 3x - 7 + 9
x^2 - 3x + 2
(x - 1)(x - 2) --> check with FOIL, First Outer Inner Last
x^2 - 2x - x + 2 --> x^2 - 3x + 2, yes

2x^2 + (5)x - 6x + 14 - 3(5)
2x^2 + 5x - 6x + 14 - 15
2x^2 - x - 1
(2x + 1)(x - 1) --> check with FOIL
2x^2 - 2x + x - 1 --> 2x^2 - x - 1, yes

x - 1 is another common factor

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