Answer:
A. (x +8) + (-4x+31)/(x^2+2x+1)
Step-by-step explanation:
When you perform long division of polynomials, the first quotient term is the ratio of the highest-degree terms in the numerator and denominator: x^3/x^2 = x.
This fact eliminates all but choices A and C.
The denominator of the remainder term is the denominator of the original expression, so will be x^2 +2x +1, as shown in choice A.
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So, simply based on a couple of facts about long division (that you learned in the early elementary grades), you can make the correct choice of answer without actually working the problem in detail.
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This is a polynomial long division problem. It is worked in virtually the same way that numerical long division problems are worked: first you find a quotient term, then you multiply that by the divisor and subtract the result from the dividend. The difference is the new dividend. These steps are identical to numerical long division.
For polynomial long division, instead of lining up the digits with the same place value, you line up the terms with the same degree of the variable.
As mentioned above, the quotient term is computed only from the highest-degree terms of dividend and divisor, so that part is actually simpler than for numerical long division.
A dividend that is of lower degree than the divisor is considered to be the remainder. As with numerical long division, it can be expressed as a fraction with the divisor as the denominator.
Numerical example: 18/7 = 2 remainder 4 or 2 4/7.
