Question

If x+2 and x-2 are factors of x3+10x2+mx+n then find the value of m and n

Answer 1

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Given

Factors = ( x + 2 ) And ( x - 2 )

If they are factors of the given polynomial, then they make the whole polynomial equals to zero.

•°•

x + 2 = 0 And x - 2 = 0

x = - 2 And x = 2

Put one by one in the given polynomial

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 )

f ( x ) = x³ + 10x² + mx + n

f ( 2 ) = ( 2 )³ + 10 ( 2 )² + m ( 2 ) + n

f ( 2 ) = 8 + 10 ( 4 ) + 2m + n

0 = 8 + 40 + 2m + n

0 = 48 + 2m + n

2m + n = 48 _________( 2 )

Add ( 1 ) and ( 2 ), we get

4m = 80

m = 20

Put this value in ( 1 )

2 ( 20 ) - n = 32

40 - n = 32

n = 40 - 32

n = 8

Hence,

Value of m = 20

And

Value of n = 8

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