Pranav can assume that the remainder of the polynomial long division of m(x) by x-5 is 6, which means m(5)=6; synthetic division will also give a remainder of 6. 5 is not a root of m(x), because for it to be the root, m(5) would need to be 0.
Step-by-step explanation:
Given that Pranav has used polynomial long division to determine that (m(x))/(x - 5) = r(x) + 6/(x - 5), we can deduce several things about the polynomial m(x).
Firstly, we know that x - 5 is a factor of m(x) minus the remainder when m(x) is divided by x - 5. In polynomial division, if the dividend m(x) is divided by divisor x - 5, the quotient is r(x) and the remainder is 6. This means that when x equals 5, m(5) should provide us with the remainder which in this case is 6. This can be expressed as m(x) = r(x) * (x - 5) + 6.
The correct statements that can be assumed are:
5 is a root of m(x) - This is incorrect because substituting x = 5 into the equation should give us the remainder, which is 6, not 0 that would indicate a root.
m(5) = 6 - This is correct because when dividing m(x) by x - 5, the remainder is 6, so when x is 5, m(5) equals 6.
Synthetic division of m(x) by x-5 would have a remainder of 6 - This is correct as it restates the initial division, just using another method to perform it.
m(x) = r(x) * (x - 5) + 6 - This is correct and reflects the equation given after the polynomial long division.