Solve The Polynomial Equation By A Divisor Of The Square Form

The division of a polynomial equation for this time only contains a divider in a linear form, namely ax + b. Next we will discuss the division of the polynomial with a square divider or ax2 + bx + c. The following is given an example of a polynomial division model.

Example 1.
(3x3 – 7x2 – 11x + 4) : (x2 – x – 2)


So the model of the division of the polynomial equation is

The first way
The form of -9x - 4 is the remainder of the division, because it cannot be divided again by x2 – x – 2. In general if the polynomial f (x) is divided into the square of ax2 + bx + c, the remainder is linear or px + q.

The Second Way

To get the remainder of the polynomial division without doing the division with the model above, we can use the remaining theorems that have been studied before, namely

F(x) = H(x) . P(x) + Sisa
F(x) = H(x) (ax2 + bx + c) + px + q

For example 1 above the calculation can be done as follows,
3x3 – 7x2 – 11x + 4 = H(x)(x2 – x – 2) + (ax + b)
3x3 – 7x2 – 11x + 4 = H(x)(x-2)(x + 1) + (ax + b)

For x = 2, then -22 = 0 + 2a + b while for x = 1 then 5 = 0 - a + b
If we do the elimination method in the two equations, we will get the values ​​a and b.

2a + b = -22
- a + b = 5    -
3a = -27
  a = - 9 and b = -4

so, the remaining share is 9x - 4. Compare the results in the First Way. Apparently, the results remain the same.

Example 2.
Determine the remaining division of the polynomial equation (2x3 + 5x2 – 7x + 3) : (x2 – 4)
Answers:

x2 – 4 = (x + 2)(x – 2)
for x = - 2, then f(-2) = 21 = - 2a + b
     x =   2, then f( 2)  = 25 =   2a  + b _
                                             - 4 = - 4a
                                               a = 1 and b = 23

so the remainder of the polynomial is x + 23,


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