# Use Remainder Theorem to solve polynomial equations

In division numbers, we often get things like the
following example. At division 17: 5 is 3 with the remaining 2.

17 = (5 x 3) + 2

The numbers are 17 as divided numbers, 5 as dividers, 3 as divides, and 2 as the remainder.

By looking at the similarities above, we can also determine the basic equation that connects the polynomial

**f(x)**which in this case is a divided element.**P(x)**as a divider,**H(x)**as a result of divide and**S**is the remainder of the division, i.e.
f(x) = P(x) . H(x) + S

for P(x) = x – k, then

f(x) = (x – k) . H(x) + S

Based on the above equation, we can mention a theorem
for the remainder of the polynomial division.

## Polynomial Remainder Theorem

If the Polynomial f(x) is divided by x - k, the
remainder is f (k).

**Proof**:

Use the equation f(x) = (x – k) . H(x) + S.

The degree S is lower than x - k, therefore S is a
constant.

For x = k, it will be obtained

f(k) = (k – k) . H(k) + S

f(k) = 0. H(k) + S

so, f(k) = S (proven).

**Example 1.**

Determine the quotient and remainder of the division of the polynomial equation 3x

^{4}– 2x^{3}+ x – 7 divided by x - 2.
Then the results obtained are 3x

^{3}+ 4x^{2}+ 8x + 17 with the remaining 27.
We try to multiply between (x - 2) with the results obtained previously, the result is the initial equation before divided (x - 2).

(x – 2)( 3x

^{3}+ 4x^{2}+ 8x + 17) + 27
3x

^{4}+ 4x^{3}+ 8x^{2}+ 17x – 6x^{3}– 8x^{2}– 16x - 34 + 27
3x

^{4}+ 4x^{3}– 6x^{3 }+ 8x^{2}– 8x^{2}+ 17x – 16x - 34 + 27
3x

^{4}– 2x^{3}+ x + 61.### The Polynomial Equation Is Divided By ax - b

Using the remainder theorem above can determine a new equation for the
polynomial f (x) divided by

**ax - b**, as follows.
f(x) = (ax – b) H(x) + S

f(x) = a(x – b/a) H(x) + S

f(x) = (x – b/a) . a H(x) + S

In the above equation, it can be shown that the remainder of the polynomial division f(x) by

**ax - b**is**f(b/a).**

**Example 2.**

Determine the quotient and remainder of the division of the polynomial equation

3x3 + 5x2 – 11x + 8 divided by 3x – 1.

f(x) = (x – 1/3) (3x

^{2}+ 6x – 9) + 5
f(x) = (x – 1/3) . 3 . (x

^{2}+ 2x – 3) + 5
f(x) = (3x – 1)( x

^{2}+ 2x – 3) + 5
Then the results obtained are x

^{2}+ 2x – 3 with the remaining 5.SUBSCRIBE TO OUR NEWSLETTER

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