When x³+ax²+4 is divided by x+1, the remainder is 6 greater than the remainder when it is divided by x-2, the value of a is
Question
When x³+ax²+4 is divided by x+1, the remainder is 6 greater than the remainder when it is divided by x-2, the value of a is
Solution
To solve this problem, we will use the Remainder Theorem which states that the remainder of a polynomial f(x) divided by (x-a) is f(a).
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According to the problem, when x³+ax²+4 is divided by x+1, the remainder is 6 greater than the remainder when it is divided by x-2. So, we can write this as:
f(-1) = f(2) + 6
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Substitute x = -1 and x = 2 into the polynomial x³+ax²+4:
(-1)³+a(-1)²+4 = (2)³+a(2)²+4 + 6
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Simplify both sides:
-1+a+4 = 8+4a+4+6
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Further simplify:
a+3 = 18+4a
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Rearrange the equation to solve for a:
a - 4a = 18 - 3
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Simplify:
-3a = 15
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Solve for a:
a = -15/3 = -5
So, the value of a is -5.
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