Question

If x + 2 and x -1 are the factors of x3 + 10x2 + mx + n, then the values of m and n are respectively

Answer 1

Answer:

m = 7    and  n = -18

Step-by-step explanation:

        Using factor theorem:

 If (x + 2) is a factor of f(x),  f(-2) = 0

⇒ (-2)³ + 10(-2)² + m(-2) + n = 0

⇒ -8 + 40 - 2m + n = 0

⇒ n - 2m = -32        ...(1)

 If (x - 1) is a factor of f(x),  f(1) = 0

⇒ (1)³ + 10(1)² + m(1) + n = 0

⇒ 1 + 10 + m + n = 0  

⇒ m + n = -11            ...(2)

On subtracting (1) from (2), we get

⇒ (m + n) - (n - 2m) = -11 - (-32)

⇒ m + 2m = 32 -11

⇒ 3m = 21

⇒ m =  7

      Substituting m is (2),  we get

               7 + n =-11       ⇒ n = -18

Answer 2

Answer :

• Now, In given question Equation is x³ + 10x² + mx + n
• And the factors of these Equation is ( x + 2 ) and ( x - 1 )

$\longrightarrow$ Hence , we can say that x + 2 = 0 and x - 1 = 0

$\longrightarrow$ Hence, x = - 2 and x = 1

$\longrightarrow$ So, First we have f( -2) = 0

• f(x) = x³ + 10x² + mx + n

• f(-2) = (-2)³ + 10(-2)² + m(-2) + n

• 0 = - 8 + 10(4) - 2m + n

• 0 = - 8 + 40 - 2m + n

• 0 = 32 - 2m + n

• 2m - n = 32 _____(1)

$\longrightarrow$ Now, second we have f(1) = 0

• f(x) = x³ + 10x² + mx + n

• f(1) = (1)³ + 10(1)² + m(1) + n

• 0 = 1 + 10 + m + n

• 0 = 11 + m + n

• m + n = - 11 _____(2)

$\longrightarrow$ Now, as we know , Substitute Method

$\longrightarrow$ So, in equation (2) and (2)

we have to add them

____

2m - n = 32

+ m + n = - 11

3m + 0 = 21

____

3m = 21

m = $\sf \frac{21}{3}$

m = $\sf \frac{\cancel{21}^{\: \: 7 × \cancel3 } }{\cancel3}$

Hence,

$\longrightarrow$ m = 7

$\longrightarrow$ So, now if we put the value of m in eq (1) or (2) we will get value of n

$\longrightarrow$ Now, I want to put value of m in eq (2)

(Note:- you can put value of m in any of these equations)

m + n = -11 ____(2)

7 + n = - 11

n = -11 - 7

$\longrightarrow$ n = - 18

I hope it helps you ❤️✔️