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Answers
EXPLANATION.
f(x) = x³ + mx² + nx + 6 has (x - 2) as a factor and leaves remainder 3 when divided by (x - 3).
As we know that,
⇒ (x - 2) is a factor of the equation.
⇒ x - 2 = 0.
⇒ x = 2.
Put the value of x = 2 in the equation, we get.
⇒ x³ + mx² + nx + 6 = 0.
⇒ (2)³ + m(2)² + n(2) + 6 = 0.
⇒ 8 + 4m + 2n + 6 = 0.
⇒ 4m + 2n + 14 = 0.
Taking 2 as common from the equation, we get.
⇒ 2[2m + n + 7] = 0.
⇒ 2m + n + 7 = 0. - - - - - (1).
Divided by (x - 3) and leaves remainder = 3.
⇒ (x - 3) = 0.
⇒ x = 3.
Put the value of x = 3 in the equation, we get.
⇒ x³ + mx² + nx + 6 = 3.
⇒ (3)³ + m(3)² + n(3) + 6 = 3.
⇒ 27 + 9m + 3n + 6 = 3.
⇒ 9m + 3n + 33 = 3.
⇒ 9m + 3n + 33 - 3 = 0.
⇒ 9m + 3n + 30 = 0.
Taking 3 as common from the equation, we get.
⇒ 3[3m + n + 10] = 0.
⇒ 3n + n + 10 = 0. - - - - - (2).
From equation (1) & (2), we get.
Subtracting equation (1) & (2), we get.
⇒ 2m + n + 7 = 0. - - - - - (1).
⇒ 3n + n + 10 = 0. - - - - - (2).
⇒ - - -
We get,
⇒ - m - 3 = 0.
⇒ - m = 3.
⇒ m = - 3.
Put the value of m = - 3 in the equation (1), we get.
⇒ 2m + n + 7 = 0. - - - - - (1).
⇒ 2(-3) + n + 7 = 0.
⇒ - 6 + n + 7 = 0.
⇒ n + 1 = 0.
⇒ n = - 1.
Values of m = - 3 and n = - 1.