SOLUTION: When ax³ + bx² + cx - 4 is divided by (x+2), the remainder is double that obtained when the expression is divided by (x+1). Show that c can have any value and find b in terms of

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Question 1158341: When ax³ + bx² + cx - 4 is divided by (x+2), the remainder is double that obtained when the expression is divided by (x+1). Show that c can have any value and find b in terms of a.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

, the remainder is double that obtained when the expression is divided by (x+1). Show that c can have any value and find b in terms of a.
ax%5E3+%2B+bx%5E2+%2B+cx+-+4 is divided by %28x%2B2%29:
..........(ax%5E2%2B%28b-+2a%29x%2B%28c-2b%2B4a%29
%28x%2B2%29|ax%5E3+%2B+bx%5E2+%2B+cx+-+4
..........ax%5E3+%2B+2ax%5E2 ............subtract
..................%28b-+2a%29x%5E2%2B+cx
.................%28b-+2a%29x%5E2%2B2%28b-+2a%29x..........subtract
............................%28c-%282b-+4a%29%29x
............................%28c-2b%2B4a%29x-+4
............................%28c-2b%2B4a%29x%2B2%28c-2b%2B4a%29.........subtract
.........................................-4-2%28c-2b%2B4a%29
.........................................-4-2c%2B4b-8a->reminder

ax%5E3+%2B+bx%5E2+%2B+cx+-+4 is divided by %28x%2B2%29:


..........(ax%5E2%2B%28b-+a%29x%2B%28c-b%2Ba%29
%28x%2B1%29|ax%5E3+%2B+bx%5E2+%2B+cx+-+4
............ax%5E3+%2B+ax%5E2 ............subtract
..................%28b-+a%29x%5E2%2B+cx
.................%28b-+a%29x%5E2%2B%28b-+a%29x..........subtract
............................%28c-%28b-+a%29%29x
............................%28c-b%2Ba%29x-+4
............................%28c-b%2Ba%29x%2B%28c-b%2Ba%29.........subtract
.........................................-4-%28c-b%2Ba%29
.........................................-4-c%2Bb-a->reminder

the remainder -4-2c%2B4b-8a is double more than remainder -4-c%2Bb-a:
-4-2c%2B4b-8a=2%28-4-c%2Bb-a%29
-4-2c%2B4b-8a=-8-2c%2B2b-2a........simplify
-4-cross%282c%29%2B4b-8a=-8-cross%282c%29%2B2b-2a-> so,this proves that +c can have any+value
-4%2B4b-8a=-8%2B2b-2a.........solve for b
4b-2b=8a-8%2B4-2a
2b=6a-4
b=3a-2