12. x^3 - 2x^2 - 5x + 6
Using synthetic division to find the remaining polynomial, we have
1 [ 1 - 2 -5 6 ]
1 -1 -6
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1 -1 -6 0
The remaining polynomial is x^2 - x - 6 which factors as
( x - 3) ( x + 2)
So....the other two roots are 3 and - 2
r = √ [ x^2 + y^2 ] = √ [ ( -2)^2 + 9^2 ] = √85
cos θ = -2 / √85 = -2√85 / 85
1a + 2b = 0
2b = -1a
b = (-1/2)a
The slope of this line is (-1/2)....this line lies in the II and IV quadrants...and since the cos < 0, then this must be a II quadrant angle so the sine must > 0
So.....find r = √ [ x^2 + y^2 ] = √ ] (-2)^2 + (1)^2 ] = √5
So....the sinθ = 1/ √5 = √5 / 5
We can use synthetic division for the first
- 2 [ -2 - 5 4 2 ]
4 2 -12
-2 -1 6 -10
The remainder is -10 so the third answer is correct
x^3 - 4x^2 + x + 26 = 0
-2 is one root
Use synthetic division to find the remaining polynomial
-2 [ 1 - 4 1 26 ]
- 2 12 -26
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1 -6 13 0
The remaining polynomial is x^2 - 6x + 13
Complete the square on x and we have
x^2 - 6x + 9 = -13 + 9
(x - 3)^2 = -4 take both roots
x - 3 = ±2i
x = 3 ± 2i are the other two roots ...... first answer
x + y = 0
y = -1x
Any point on this line in quadrant II will have the coordinates (-a, a)....and this will be the terminal point on θ
So tan θ = a / -a = -1
(N - 6) / 4 + 10 = 30 subtract 10 from both sides
(N - 6) / 4 = 20 multiply both sides by 4
N - 6 = 80 add 6 to both sides
N = 86
(3p^2 + 5pq-q^2 ) + (p^2 + 3pq - 2q^2 ) =
3p^2 + p^2 + 5pq + 3pq - q^2 - 2q^2 =
4p^2 + 8pq - 3q^2
27x^3 + 64 this is a difference of two cubes
( 3x + 4) ( ( 3x)^2 - (3x * 4) + 4^2) =
(3x + 4) ( 9x^2 - 12x + 16)
JKLM ~ WXYZ
So JK ~ WX
So the scale factor = WX / JK
Scale Factor = 18 / 12 = 3 / 2 = 1.5
First Position Number + _____ + _______
1 18 1
1 17 2
1 16 3
1 15 4
1 14 5
1 13 6
1 12 7
1 11 8
1 10 9
1 9 10 (repeats from here)
2 16 2
2 15 3
2 14 4
2 13 5
2 12 6
2 11 7
2 10 8
2 9 9
2 8 10 ( repeats from here)
3 14 3
3 13 4
3 12 5
3 11 6
3 10 7
3 9 8
3 8 9 (repeats from here )
4 12 4
4 11 5
4 10 6
4 9 7
4 8 8
4 7 9 (repeats from here )
5 10 5
5 9 6
5 8 7
5 7 8 (repeats from here )
6 8 6
6 7 7
6 6 8 (repeats from here )
Everything else from this point forward is a repeat
So we have 9 + 8 + 6 + 5 + 3 + 2 = 17 + 11 + 5 = 33 possibilities
We can find the scale factor as follows :
(8,2) = [ ( -8, 2) - (4,2) ] * Scale Factor + (4,2)
(8,2) = [ -12, 0] *Scale Factor + (4, 2)
(8, 2) - (4, 2) = [ -12, 0 ] * Scale Factor
(4, 0) = [ -12, 0 ] * Scale Factor
Scale Factor = (4/ -12) = -1/3
