(x-2) is a factor of the expression x^3+ax^2+bx+6 when this expression is divided by (x-3) it leaves the remainder 3. Find the values of a and b hence factorise the expression.

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Solution

As (x – 2) is a factor of f(x) = x³ + ax² + bx + 6,

Therefore, f(2) = 0

Or, (2)³ + a(2)² + b(2) + 6 = 0

Or, 8 + 4a + 2b + 6 = 0

Or, 4a + 2b = – 14

Or, 2a + b = – 7 ——————– (i)

On dividing f(x) by (x – 3) the remainder is 3

Therefore, f(3) = 3

Or, (3)³ + a(3)² + b(3) + 6 = 3

Or, 27 + 9a + 3b + 6 = 3

Or, 9a + 3b = – 30

Or, 3a + b = – 10 ——————– (ii)

Subtracting (ii) from (i) we get a = – 3

Substituting value of a = – 3 in (i) we get

2×(– 3) + b = – 7

Or, – 6 + b = – 7

Or, b = – 1.

Therefore, a = – 3, b = – 1

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