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Which of the following is the value of mm for which the polynomial x4+10x3+25x2+15x+mx^4 + 10x^3 + 25x^2 + 15x + m is exactly divisible by x+7x+7?

Solution

✅ Correct Option: 1

If polynomial is divisible by (x+7)(x + 7), then x=−7x = -7 should be a root. [Because x+7=0]x+7=0]

Substituting x=−7x = -7:

(−7)4+10(−7)3+25(−7)2+15(−7)+m(-7)^4 + 10(-7)^3 + 25(-7)^2 + 15(-7) + m

=2401−3430+1225−105+m= 2401 - 3430 + 1225 - 105 + m

=91+m= 91 + m

For polynomial to be exactly divisible by (x+7)(x + 7), this sum should be zero:

91+m=091 + m = 0

m=−91m = -91

Therefore, m=−91m = -91

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