Question

The expression ax³-9x²+bx+3a is exactly divisible by x²-5x+6 . Find the value of a and b

Answer 1

Note that, since x^2 - 5x + 6

= (x - 3)(x - 2), ax^3 - 9x^2 + bx + 3a is divisible by both x - 3 and x - 2.

By the Factor Theorem,

if x - c is a factor of f(x), then f(c) = 0.

Thus, if f(x) = ax^3 - 9x^2 + bx + 3a,

we have that:

f(3) = 0 and f(2) = 0

==> a(3)^3 - 9(3)^2 + b(3) + 3a = 0 and a(2)^3 - 9(2)^2 + b(2) + 3a = 0

==> 30a + 3b - 81 = 0 and 10a + 2b - 36 = 0.

Solving for a and b simultaneously yields:

a = 9/5 and b = 6

Answer 2

Answer:

a=1.384

b=3.461

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