Question

the polynomials x³+mx²-x+n and x³+nx²-5x+3m have a common factor of x+2. Find m and n​

Answer 1

GIVEN :–

• The polynomials x³+mx²-x+n = 0 and x³+nx²-5x+3m = 0 have a common factor of x+2.

TO FIND :–

• Value of 'm' & 'n' = ?

SOLUTION :–

• According to the question x = -2 will satisfy to both equations.

• So that –

⇛ (-2)³ + m(-2)² -(-2) + n = 0

⇛ - 8 + 4m + 2 + n = 0

⇛ - 6 + 4m + n = 0

⇛ n = 6 - 4m ________eq.(1)

• And –

⇛ (-2)³ + n(-2)² - 5(-2) + 3m = 0

⇛ - 8 + 4n + 10 + 3m = 0

⇛ 4n + 3m + 2 = 0

• Using eq.(1) –

⇛ 4(6 - 4m) + 3m +2 = 0

⇛ 24 - 16m + 3m + 2 = 0

⇛ 26 - 13m = 0

⇛ 13m = 26

⇛ m = 2

• Again using eq.(1) –

⇛ n = 6 - 4(2)

⇛ n = 6 - 8

⇛ n = - 2

• Hence , The values of 'm' is 2 and 'n' is -2 .

Answer 2

Answer:

Let p(x)=x

3

+mx

2

−x+6 as (x−2) is a factor of p(x)

⇒p(2)=0⇒8+4m−2+6=0⇒4m=−12⇒m=−3

x−3)x 3 −3x 2 −x+6(x 2 −1 x 3 −3x 2−x+6 +x+3

3=n

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