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1 answer:
Factor the following:
4 p (4 p - 1) - 3 (4 p - 1)^2
Factor 4 p - 1 from 4 p (4 p - 1) - 3 (4 p - 1)^2:
(4 p - 1) (4 p - 3 (4 p - 1))
-3 (4 p - 1) = 3 - 12 p:
(4 p - 1) (4 p + 3 - 12 p)
Grouping like terms, 4 p - 12 p + 3 = 3 + (4 p - 12 p):
(4 p - 1) (3 + (4 p - 12 p))
4 p - 12 p = -8 p:
Answer: (4 p - 1) (-8 p + 3)
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Factor the following:
m (m + n) - 3 m (m + n)^2
Factor m + n out of m (m + n) - 3 m (m + n)^2, resulting in (m + n) (m - 3 m (m + n)^(2 - 1)):
(m + n) (m - 3 m (m + n)^(2 - 1))
2 - 1 = 1:
(m + n) (m - 3 m (m + n))
Factor m out of m - 3 m (m + n), resulting in m (1 - 3 (m + n)):
m (1 - 3 (m + n)) (m + n)
-3 (m + n) = -3 m - 3 n:
Answer: m (m + n) (-3 m - 3 n + 1)
___________________________________________________________Answer : -(-6 + p^2) (-1 - 6 q + p^2 q)
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Factor the following:
6 u^2 (2 u - 5)^2 - 12 u^2 (2 u - 5) (u + 5)
Factor u^2 out of 6 u^2 (2 u - 5)^2 - 12 u^2 (2 u - 5) (u + 5), resulting in u^2 (6 (2 u - 5)^2 - 12 (2 u - 5) (u + 5)):
u^2 (6 (2 u - 5)^2 - 12 (2 u - 5) (u + 5))
(2 u - 5) (2 u - 5) = (2 u) (2 u) + (2 u) (-5) + (-5) (2 u) + (-5) (-5) = 4 u^2 - 10 u - 10 u + 25 = 4 u^2 - 20 u + 25:
u^2 (6 4 u^2 - 20 u + 25 - 12 (2 u - 5) (u + 5))
6 (4 u^2 - 20 u + 25) = 24 u^2 - 120 u + 150:
u^2 (24 u^2 - 120 u + 150 - 12 (2 u - 5) (u + 5))
(u + 5) (2 u - 5) = (u) (2 u) + (u) (-5) + (5) (2 u) + (5) (-5) = 2 u^2 - 5 u + 10 u - 25 = 2 u^2 + 5 u - 25:u^2 (150 - 120 u + 24 u^2 - 122 u^2 + 5 u - 25)
-12 (2 u^2 + 5 u - 25) = -24 u^2 - 60 u + 300:
u^2 (-24 u^2 - 60 u + 300 + 24 u^2 - 120 u + 150)
Grouping like terms, 24 u^2 - 24 u^2 - 60 u - 120 u + 150 + 300 = (-120 u - 60 u) + (24 u^2 - 24 u^2) + (150 + 300):
u^2 ((-120 u - 60 u) + (24 u^2 - 24 u^2) + (150 + 300))
-120 u - 60 u = -180 u:
u^2 (-180 u + (24 u^2 - 24 u^2) + (150 + 300))
150 + 300 = 450:
u^2 (-180 u + (24 u^2 - 24 u^2) + 450)
24 u^2 - 24 u^2 = 0:
u^2 (450 - 180 u)
Factor 90 out of 450 - 180 u:
Answer: u^2×90 (5 - 2 u)
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4 p (4 p - 1) - 3 (4 p - 1)^2
Factor 4 p - 1 from 4 p (4 p - 1) - 3 (4 p - 1)^2:
(4 p - 1) (4 p - 3 (4 p - 1))
-3 (4 p - 1) = 3 - 12 p:
(4 p - 1) (4 p + 3 - 12 p)
Grouping like terms, 4 p - 12 p + 3 = 3 + (4 p - 12 p):
(4 p - 1) (3 + (4 p - 12 p))
4 p - 12 p = -8 p:
Answer: (4 p - 1) (-8 p + 3)
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Factor the following:
m (m + n) - 3 m (m + n)^2
Factor m + n out of m (m + n) - 3 m (m + n)^2, resulting in (m + n) (m - 3 m (m + n)^(2 - 1)):
(m + n) (m - 3 m (m + n)^(2 - 1))
2 - 1 = 1:
(m + n) (m - 3 m (m + n))
Factor m out of m - 3 m (m + n), resulting in m (1 - 3 (m + n)):
m (1 - 3 (m + n)) (m + n)
-3 (m + n) = -3 m - 3 n:
Answer: m (m + n) (-3 m - 3 n + 1)
___________________________________________________________Answer : -(-6 + p^2) (-1 - 6 q + p^2 q)
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Factor the following:
6 u^2 (2 u - 5)^2 - 12 u^2 (2 u - 5) (u + 5)
Factor u^2 out of 6 u^2 (2 u - 5)^2 - 12 u^2 (2 u - 5) (u + 5), resulting in u^2 (6 (2 u - 5)^2 - 12 (2 u - 5) (u + 5)):
u^2 (6 (2 u - 5)^2 - 12 (2 u - 5) (u + 5))
(2 u - 5) (2 u - 5) = (2 u) (2 u) + (2 u) (-5) + (-5) (2 u) + (-5) (-5) = 4 u^2 - 10 u - 10 u + 25 = 4 u^2 - 20 u + 25:
u^2 (6 4 u^2 - 20 u + 25 - 12 (2 u - 5) (u + 5))
6 (4 u^2 - 20 u + 25) = 24 u^2 - 120 u + 150:
u^2 (24 u^2 - 120 u + 150 - 12 (2 u - 5) (u + 5))
(u + 5) (2 u - 5) = (u) (2 u) + (u) (-5) + (5) (2 u) + (5) (-5) = 2 u^2 - 5 u + 10 u - 25 = 2 u^2 + 5 u - 25:u^2 (150 - 120 u + 24 u^2 - 122 u^2 + 5 u - 25)
-12 (2 u^2 + 5 u - 25) = -24 u^2 - 60 u + 300:
u^2 (-24 u^2 - 60 u + 300 + 24 u^2 - 120 u + 150)
Grouping like terms, 24 u^2 - 24 u^2 - 60 u - 120 u + 150 + 300 = (-120 u - 60 u) + (24 u^2 - 24 u^2) + (150 + 300):
u^2 ((-120 u - 60 u) + (24 u^2 - 24 u^2) + (150 + 300))
-120 u - 60 u = -180 u:
u^2 (-180 u + (24 u^2 - 24 u^2) + (150 + 300))
150 + 300 = 450:
u^2 (-180 u + (24 u^2 - 24 u^2) + 450)
24 u^2 - 24 u^2 = 0:
u^2 (450 - 180 u)
Factor 90 out of 450 - 180 u:
Answer: u^2×90 (5 - 2 u)
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So essentially, this is a cubic function where T(x) = y, and:
In order to find the vertices, aka "turning points", we need to see around what area the graph will curve quickly from bottom-left to top-right. If the value being cubed (in the parentheses) is around -8, y will be close to 0, since -8+7=-1. (-2)^3 = -8. So x+5=-2, x=-7.
At x=-6, y = (-6+5)^3+7 = (-1)^3+7
y = -1+7 = 6
At x=-5, y = 0+7 = 7
Above x=-4, the y jumps rapidly above 10, and below x=-9, the y jumps rapidly below -50
What does all that mean???
That between x = -9 and x = -4 (x = -4,-5,-6,-7,-8,-9), the temperature (T(x) or y) does not change very much. Therefore, around those temperatures is a turning point, where the phase shift occurs from solid/liquid.
In order to find the vertices, aka "turning points", we need to see around what area the graph will curve quickly from bottom-left to top-right. If the value being cubed (in the parentheses) is around -8, y will be close to 0, since -8+7=-1. (-2)^3 = -8. So x+5=-2, x=-7.
At x=-6, y = (-6+5)^3+7 = (-1)^3+7
y = -1+7 = 6
At x=-5, y = 0+7 = 7
Above x=-4, the y jumps rapidly above 10, and below x=-9, the y jumps rapidly below -50
What does all that mean???
That between x = -9 and x = -4 (x = -4,-5,-6,-7,-8,-9), the temperature (T(x) or y) does not change very much. Therefore, around those temperatures is a turning point, where the phase shift occurs from solid/liquid.