IF (X - 2) IS A FACTOR OF THE EXPRESSION 2X^3 + AX^2 + BX - 14 AND WHEN THE EXPRESSION IS DIVIDED BY (X - 3), IT LEAVES A REMINADER 52. FIND THE VALUES OF A AND B ?
Answers
Explanation :
Given :-
- (x - 2) is a factor of the expression 2x³ + ax² + bx - 14.
- When the expression is divided by (x - 3), it leaves a remainder 52.
To find :-
- Values of a and b.
Solution :-
Let f(x) = 2x³ + ax² + bx - 14.
Since (x + 2) is a factor of f(x), we must have f(2) = 0.
Now, f(2) = 0:
⇒ [2 × 2³ + a × 2² + b × 2 - 14] = 0
⇒ [2 × 8 + a × 4 + 2b - 14] = 0
⇒ [16 + 4a + 2b - 14] = 0
⇒ [4a + 2b + 2] = 0
⇒ 2a + b = -1 - (1)
Again, by remainder theorem, on dividing f(x) by (x - 3), we have f(3) as remainder.
∴ f(3) = 52
⇒ [2 × 3³ + a × 3² + b × 3 - 14] = 52
⇒ [2 × 27 + a × 9 + 3b - 14] = 52
⇒ [54 + 9a + 3b - 14] = 52
⇒ [40 + 9a + 3b] = 52
⇒ 9a + 3b = 12
⇒ 3a + b = 4 - (2)
Solving (1) and (2) :
⇒ 2a + b = -1
⇒ 3a + b = 4
- a = 5
⇒ 2a + b = -1
⇒ 2(5) + b = -1
- 10 + b = -1
- b = -11
∴ The values of a and b are 5 and -11 respectively.
EXPLANATION.
⇒ (x - 2) is a factor of the equation.
⇒ 2x³ + ax² + bx - 14.
As we know that,
⇒ (x - 2) is a factor of equation.
⇒ x - 2 = 0.
⇒ x = 2.
Put the value of x = 2 in the equation, we get.
⇒ 2(2)³ + a(2)² + b(2) - 14 = 0.
⇒ 16 + 4a + 2b - 14 = 0.
⇒ 4a + 2b + 2 = 0.
⇒ 2a + b + 1 = 0. - - - - - (1).
When the expression is divided by (x - 3) it leaves remainder 52.
As we know that,
⇒ (x - 3) is a factor of the equation.
⇒ x - 3 = 0.
⇒ x = 3.
Now, we can write equation as,
⇒ 2x³ + ax² + bx - 14 = 52.
⇒ 2(3)³ + a(3)² + b(3) - 14 = 52.
⇒ 54 + 9a + 3b - 14 = 52.
⇒ 9a + 3b + 40 = 52.
⇒ 9a + 3b + 40 - 52.
⇒ 9a + 3b - 12 = 0. - - - - - (2).
⇒ 3a + b - 4 = 0. - - - - - (2).
From equation (1) and (2), we get.
⇒ 2a + b + 1 = 0. - - - - - (1).
⇒ 3a + b - 4 = 0. - - - - - (2).
Subtract both the equation, we get.
⇒ 2a + b + 1 = 0. - - - - - (1).
⇒ 3a + b - 4 = 0. - - - - - (2).
⇒ - - +
We get,
⇒ - a + 5 = 0.
⇒ a = 5.
Put the values of a = 5 in the equation (1), we get.
⇒ 2a + b + 1 = 0.
⇒ 2(5) + b + 1 = 0.
⇒ 10 + b + 1 = 0.
⇒ b + 11 = 0.
⇒ b = - 11.
Values of a = 5 and b = - 11.