If (x-2) is a factor of the expression 2xcube+axsquare+bx-14 and when the expression is divided by (x-3), it leaves a remainder 52, find the values a and b.
Answers
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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.