The polynomial 2x³-ax²+bx+6 is exactly divisible by x²-5x+6. Find the value of a and b.

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