if x+1 is a factor of 2x^3+ax^2+2bx+1,then find the value of a and b given that 2a-3b=4
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x+1 is a factor of 2x³+ax²+2bx+1
x+1 = 0
x = -1
→ 2(-1)³+a(-1)²+2b(-1)+1 = 0
→ 2(-1) + a(1) - 2b + 1 = 0
→ -2+a-2b+1 = 0
→ a-2b-1 = 0
→ a-2b = 1
Multiply it by 2,
2(a-2b) = 2(1)
2a-4b = 2 -----(1)
Given,
2a-3b = 0 ------(2)
(1) - (2)
2a-4b = 2
-{2a-3b = 0}
––––––––
-b = 2
b = -2
2a - 3(-2) = 0
2a+6 = 0
2a = -6
a = -6/2
a = -3
Therefore, a = -3 and b = -2
x+1 = 0
x = -1
→ 2(-1)³+a(-1)²+2b(-1)+1 = 0
→ 2(-1) + a(1) - 2b + 1 = 0
→ -2+a-2b+1 = 0
→ a-2b-1 = 0
→ a-2b = 1
Multiply it by 2,
2(a-2b) = 2(1)
2a-4b = 2 -----(1)
Given,
2a-3b = 0 ------(2)
(1) - (2)
2a-4b = 2
-{2a-3b = 0}
––––––––
-b = 2
b = -2
2a - 3(-2) = 0
2a+6 = 0
2a = -6
a = -6/2
a = -3
Therefore, a = -3 and b = -2
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