Question

GIVEN THAT (X + 2) AND (X + 3) ARE THE FACTORS OF 2X^3 + AX^2 + 7X - B. DETERMINE THE VALUES OF A AND B?

Answer 1

Explanation :

Given :-

• (x + 2) and (x + 3) are the factors of 2x³ + ax² + 7x - b.

To find :-

• Values of a and b.

Solution :-

Let f(x) = 2x³ + ax² + 7x - b.

Since (x + 2) is a factor of f(x), we must have f(-2) = 0.

Again, since (x + 3) is a factor of f(x), we must have f(-3) = 0.

Now, f(-2) = 0

⇒ [2 × (-2)³ + a × (2)² + 7 × (-2) - b] = 0

⇒ [2 × (-8) + a × 4 + (-14) - b] = 0

⇒ [-16 + 4a - 14 - b] = 0

⇒ [-30 + 4a - b] = 0

⇒ 4a - b = 30  - (1)

Now, f(-3) = 0

⇒ [2 × (-3)³ + a × (-3)² + 7(-3) - b = 0

⇒ [2 × (-27) + a × (9) + (-21) - b = 0

⇒ [-54 + 9a - 21 - b] = 0

⇒ 9a - b = 75  - (2)

Solving (1) and (2) :

⇒ 4a - b = 30

⇒ 9a - b = 75

___________

• 5a = 45
• a = 9

⇒ 9a - b = 75

⇒ 9(9) - b = 75

• 81 - b = 75
• b = 6

∴ The values of a and b are 9 and 6 respectively.

Answer 2

EXPLANATION.

⇒ (x + 2) and (x + 3) are the factors.

⇒ 2x³ + ax² + 7x - b.

As we know that,

⇒ (x + 2) is a factors of equation.

⇒ x + 2 = 0.

⇒ x = - 2.

Put the value of x = - 2 in the equation, we get.

⇒ 2(-2)³ + a(-2)² + 7(-2) - b = 0.

⇒ - 16 + 4a - 14 - b = 0.

⇒ 4a - b - 30 = 0.

⇒ 4a - 30 = b. - - - - - (1).

⇒ (x + 3) is a factors of the equation.

⇒ x + 3 = 0.

⇒ x = - 3.

Put the value of x = - 3 in the equation, we get.

⇒ 2(-3)³ + a(-3)² + 7(-3) - b = 0.

⇒ - 54 + 9a - 21 - b = 0.

⇒ 9a - b - 75 = 0.

⇒ 9a - 75 = b. - - - - - (2).

From equation (1) and (2), we get.

We can write equation as,

⇒ 4a - 30 = 9a - 75.

⇒ - 30 + 75 = 9a - 4a.

⇒ 45 = 5a.

⇒ a = 9.

Put the values of a = 9 in the equation, we get.

⇒ 4a - 30 = b.

⇒ b = 4(9) - 30.

⇒ b = 36 - 30.

⇒ b = 6.

Values of a = 9 and b = 6.