If (x - 1) is a factor......then x = 1 is a root.....therefore
4(1)^4 - 7(1)^3 + m(1)^2 + n(1) + 6 = 0
4 - 7 + 6 + m + n = 0
3 + m + n = 0
m + n = -3 (1)
And by the Remainder Theorem
P(-1) = 30.....so......
4(-1)^4 - 7(-1)^3 + m(-1)^2 + n(-1) + 6 = 30
4 + 7 + 6 + m - n = 30
17 + m - n = 30
m - n = 13 (2)
Add (1) and (2) and we get that
2m = 10
m = 5
And m + n = -3 ⇒ n = -8
a + b = 7
Square both sides
a^2 + 2ab + b^2 = 49
a^2 + b^2 = 49 - 2ab
a^3 + b^3 = (a + b) (a^2 - ab + b^2) = 45
(a + b) ( a^2 + b^2 - ab) = 45
(7) ( 49 - 2ab - ab) = 45
49 - 3ab = 45/7
49 - 45/7 = 3ab
298/7 = 3ab
298 / 21 = ab
And
1/a + 1/b = (a + b) / ab = 7 / (298/21) = 147 / 298
x coordinate = (2+ 6) / 2 = 4
y coordinate = ( 5 + 3) / 2 = 4
4 * 4 = ?????
b + s = 14
5b + 10s = 80
2x - 5 < -x - 12
2x + x < -12 + 5
3x < -7
x < -7/3
( -inf , -7/3)
We have similar 3D figures
The scale factor of the large bucket to the smaller = 2
Volume of larger bucket =
Volume of smaller bucket * (scale factor )^ 3 =
850 (2)^3 =
6800 cm^3 (as EP found)
Let one root = a and the other be (7/2)a
By Vieta :
a + (7/2)a = -1
(9/2)a = -1
a = -1/(9/2) = -2/9
(7/2)a = (7/2)(-2/9) = -7/9
Also, byVieta :
a * (7/2)a = k
(-2/9) (-7/9) = k = 14/81
Graphical solution : https://www.desmos.com/calculator/jxu8ptf7ql
Result = 24
Proof Let a ≈ 20 Let a = 21
abs( ≈20- 20) = 0 abs ( 21 - 20 ) = 1
abs(6 - 0) = 6 abs (6 - 1) = 5
abs ( 2 - ≈20 - 6) = abs ( ≈ -24) ≈ 24 abs ( 2 - 21- 5) = abs ( -24) = 24
arc LPM = 360 - 72 = 288°
Need to be careful about dividing out possible solutions.........
2cos^2x = cos x
2cos^2 x - cos x = 0
cos x ( 2cos x - 1) = 0
Set each factor to 0 , and solve for x
cos x = 0 x = pi/2 and x = 3pi/2 2cos x - 1 = 0
2cosx = 1
cosx = 1/2 x = pi/3 and x = 5pi/3