Answer:
a = 9
b = 7
Explanation:
For the given polynomial 2x³ - ax² + bx + 6 to be exactly divisible by x² - 5x + 6, it means it must also be divisible by the individual factors of x² - 5x + 6.
So let's factorize x² - 5x + 6.
Factorizing it gives us;
(x - 2)(x - 3)
Thus means that the given polynomial should be exactly divisible by either (x - 2) or (x - 3).
This means that at x = 2 or 3, the polynomial will equal zero.
Thus;
f(2) = 2(2)³ - a(2)² + b(2) + 6 = 0
16 - 4a + 2b + 6 = 0
Adding and Rearranging we have;
4a - 2b = 22
Divide through by 2 to get;
2a - b = 11 - - - (eq 1)
Similarly;
f(3) = 2(3)³ - a(3)² + b(3) + 6 = 0
54 - 9a + 3b + 6 = 0
Adding and Rearranging we have;
9a - 3b = 60
Divide through by 3 to get;
3a - b = 20 - - - (eq 2)
Subtract eq 1 from eq 2 to get;
3a - 2a - b - (-b) = 20 - 11
a = 9
Put 9 for a in eq 2 to get;
3(9) - b = 20
27 - b = 20
b = 27 - 20
b = 7
