# 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 ax

^{2}+ bx + c. The following is given an example of a polynomial division model.
Example 1.

(3x

^{3}– 7x^{2}– 11x + 4) : (x^{2}– 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 x

^{2}– x – 2. In general if the polynomial f (x) is divided into the square of**ax**, the remainder is linear or^{2}+ bx + c**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) (ax

^{2}+ bx + c) + px + q
For example 1 above the calculation can be done as follows,

3x

^{3}– 7x^{2}– 11x + 4 = H(x)(x^{2}– x – 2) + (ax + b)
3x

^{3}– 7x^{2}– 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 (2x

^{3}+ 5x^{2}– 7x + 3) : (x^{2}– 4)
Answers:

x

^{2}– 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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