✪ Question :-

If 2x^3 + ax^2 - bx -15 has (2x+3) as a factor and leaves a remainder -5 when divided by (x-1) find the value of a and B ??

✰ Formula and CONCEPT used :-

➳ The remainder THEOREM states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a).

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❁ Solution :-

It has been given that (2x+3) is a factor of given polynomial , so, we can say that,

⟿ 2x + 3 = 0

⟿ 2x = (-3)

⟿ x = (-3/2)

At (-3/2) the Remainder will be zero.

So, putting we get,

➺ f(x) = 2x^3 + ax^2 - bx -15 = 0

➺ f(-3/2) = 2*(-3/2)³ + a(-3/2)² - b(-3/2) - 15 = 0

➺ f(-3/2) = 2*(-27/8) + a(9/4) + 3b/2 -15 = 0

➺ f(-3/2) = -54/8 + 9a/4 + 3b/2 - 15 = 0

→ 9a/4 + 3b/2 = 15 + 54/8

➺ (9a + 6b)/4 = (120 + 54)/8

→ 2(9a + 6b) = 174

→ 9a + 6b = 87

→ 3(3a + 2b) = 87

→ 3a + 2b = 29 ------------ Equation (1)

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Now, it has been given that , the given polynomial gives remainder (-5) when divided by (x-1).

So,

→ f(1) = 2(1)³ + a(1)² -b*1 - 15 = (-5)

→ f(1) = 2 + a - b - 15 = (-5)

→ a - b = (-5) + 15 - 2

→ a - b = 8 ---------------- Equation (2)

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Now, Multiplying Equation (2) by 2 and than adding In Equation (1) we get,

➪ (3a + 2b ) + 2(a - b) = 29 + 2*8

➪ 3a + 2b + 2A - 2b = 29 + 16

➪ 5a = 45

➪ a = 9

Putting this value in Equation (2) now, we get,

☛ b = a - 8 = 9 - 8 = 1 .

Hence, value of a is 9 and b is 1.