Question

if x^3+mx^2-x+b has (x-2) as a factor, and leaves a remainder n when divided by(x-3) , find the value of m and n​

Answer 1

Correct question:

if x^3+mx^2-x+6 has (x-2) as a factor, and leaves a remainder n when divided by(x-3) , find the value of m and n

Answer:

f(x) = x^3+mx^2-x+6

As one (x-2) is one of its factors,

x - 2 = 0

=> x = 2

Now substituting the value if x in f(x) we have,

f(2) = 2^3 + m × 2^2 - 4 + 6

= 8 + 4m + -4 + 6 = 0

= 8 + 4 + 4m = 0

= 12 + m = -12

= m = -12/4

=> m = -3

Hence, the polynomial is  x^3-3x^2-x+6

Now, when the polynomial is divided by (x-3)

x - 3 = 0

=> x = 3

Substituting the value of x = 3 we get

f(3) = 3^3 - m(3)^2 - 3 + 6

n = 27 + 9m + 4

n = 27 - 9(3) + 4

n = 27 - 27 + 4

=> n = 4

Answer 2

$\large\underline\mathrm{Solution}$

$\implies$ f(x) = x³ + mx² - x + 6

As a one factor (x - 2)

$\implies$ x - 2 = 0

$\implies$ x = 2

Now, Substitution the value of x in f(x).

$\implies$ f(x) => x³ + mx² - x + 6

$\implies$ f(2) => 2³ + m(2)² - 2 + 6

$\implies$ f(2) => 8 + 4m + 4 = 0

$\implies$ f(2) => 12 + 4m = 0

$\implies$ f(2) => 4m = -12

$\implies$ f(2) => m = -12/4

$\implies$ f(2) => m = -3

Thus, the polynomial is x³ - 3x² - x + 6.

As a one factor (x - 3)

$\implies$ x - 3 = 0

$\implies$ x = 3

Now, Substitution the value of x in f(x).

$\implies$ f(x) => x³ + mx² - x + 6

$\implies$ f(3) => 2³ + m(3)² - 3 + 6

$\implies$ f(3) => 27 + 9m + 3 = 0

$\implies$ f(3) => 30 + 9m = 0

$\implies$ f(3) => 9m = 30

$\implies$ f(3) => 9(-3) + 30

$\implies$ f(3) => -27 + 30

$\implies$ f(3) => 3

$\implies$ n = 3.