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Answer 1 · 2017-09-05T10:23:16

There are two possible values:

# m=-3/2# or #m= 2 #

We use the remainder theorem:

The remainder of the division of a polynomial #f(x)# by a linear factor #x-a# is #f(a)#

Consider the first polynomial:

# P(x)=x^3 +4x^2 - 2x +1 #

Then by the Remainder Theorem, if we divide #P(x)# by #x-m# then the remainder, #r_p#, is:

# r_p = P(m) = m^3 +4m^2 - 2m +1 #

Similarly for the second polynomial:

# Q(x) = x^3 +2x^2 - x + 7 #

If we divide #Q(x)# by #x-m# then the remainder, #r_q#, is:

# r_q = Q(m) = m^3 +2m^2 - m + 7 #

We are given that the remainders are the same:

# :. r_p = r_q #
# :. m^3 +4m^2 - 2m +1 = m^3 +2m^2 - m + 7 #
# :. 2m^2 - m -6 = 0 #
# :. (m-2)(2m+3) = 0 #
# :. m=-3/2, 2 #