If we divide the polynomial x4 + 4x3 + 2x2 + x + 4 by x2 + 3x, what will be the remainder?
Solution:
In this problem, the divisor is x2 + 3x which is not of the form x + a or x - a. Also the equation that needed to be divided i.e. x4 + 4x3 + 2x2 + x + 4 has a constant term. To find the solution the long division shall be the most appropriate method in this case as the divisor has all terms in variable x and has no constant.
Step1: Multiplying x2 + 3x by x2 and subtracting from x4 + 4x3 we have
x4 + 4x3
-x4 + 3x3
------------
0 + x3
Step2: The new dividend becomes x3 + 2x2 once again divided by x2 + 3x. This time multiply the divisor by x and subtract it from x3 + 2x2
x3 + 2x2
-x3 + 3x2
-----------
0 - x2
Step3: The new dividend becomes -x2 + x. This time multiply the divisor x2 + 3x by -1 and subtract from the new dividend as shown:
-x2 + x
-(-x2 - 3x)
-----------
0 + 4x
The new dividend becomes 4x + 4 and this cannot be further divided and hence becomes the remainder. The quotient therefore becomes x2 + x - 1.
If we divide the polynomial x4 + 4x3 + 2x2 + x + 4 by x2 + 3x, what will be the remainder?
Summary:
If we divide the polynomial x4 + 4x3 + 2x2 + x + 4 by x2 + 3x, then the remainder is 4(x+1). Therefore simple long division was the most appropriate method. The solution could be expressed as (x2 + 3x)(x2 + x -1) + (4x + 4). The quotient being (x2 + x -1) and the remainder being 4x+4 or 4(x+1).
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