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Example 6. Find the value of a and b so that the polynomial x³ + ax² + bx - 6 is completely divisible by x² - 4x + 3.

Solution: x² - 4x + 3 = x² - 3x - x + 3

= x(x - 3) -1(x - 3)

(x - 3) (x - 1)

Let f(x) = x³ + ax + bx - 6

as x - 3 is a factor of f(x)

f(3) = 0

i.e. 3³ + a(3)² + b(3) - 6 = 0

Or, 27 + 9a + 3b - 6 = 0

Or, 9a + 3b = -21

Or, 3a + b = -7 -------------(i)

Also, x - 1 is a factor of f(x) f(1) = 0

i. e. 1³ + a(1)² + b(1) - 6 = 0

Or, a + b = 5 ---------------(ii)

Subtracting (i) from (ii) we get -2a = 12 a = -6.

Putting a = -6 in (ii) we get

b = 5 - b = 5 - (-6)

= 5 + 6 = 11

a = -6, b = 11