# REMAINDER THEOREM AND FACTOR THEOREM

## About "Remainder theorem and Factor theorem"

Remainder theorem and Factor theorem :

In this section, we are going to see remainder and factor theorem.

Remainder theorem :

If a polynomial P(x) is divided by (x-a), the remainder is P(a).

Using division algorithm, we have

P(x)  =  Q(x)(x-a) + P(a)

Here, Q(x) is the quotient when P(x) is divided by (x-a).

Factor theorem :

A polynomial P(x) would have a factor (x-a), if and only if P(a)  =  0.

## Remainder theorem and Factor theorem - Examples

Example 1 :

Using Remainder theorem, find the remainder when the polynomial 3x³- 2x² + 6x - 7 is divided by (x-2).

Solution :

Step 1 :

Let P(x)  =  3x³- 2x² + 6x - 7.

If x - 2 = 0, then x  =  2.

Step 2 :

To know the remainder when P(x) divided by (x-2), aaaaa     plug x = 2 in P(x).

Remainder  =  P(2)

Remainder  =  3(2)³- 2(2)² + 6(2) - 7

Remainder  =  24 - 8 + 12 - 7

Remainder  =  21

Example 2 :

Using Remainder theorem, find the remainder when the polynomial  7x - x²- 3x + 9 is divided by (x-6).

Solution :

Step 1 :

Let P(x)  =  7x - x²- 3x + 9

If x - 6 = 0, then x  =  6.

Step 2 :

To know the remainder when P(x) divided by (x-6), aaaaa     plug x = 6 in P(x).

Remainder  =  P(6)

Remainder  =  7(6)⁴ - (6)² - 3(6) + 9

Remainder  =  9072 - 36 - 18 + 9

Remainder  =  9027

Example 3 :

Using Remainder theorem, find the remainder when the polynomial  x³+ 3x² - 5x + 2 is divided by (x+5).

Solution :

Step 1 :

Let P(x)  =  x³+ 3x² - 5x + 2

If x + 5 = 0, then x  =  - 5.

Step 2 :

To know the remainder when P(x) divided by (x+5), aaaaa     plug x = - 5 in P(x).

Remainder  =  P(-5)

Remainder  =   (-5)³+ 3(-5)² - 5(-5) + 2

Remainder  =  -125 + 3(25) + 25 + 2

Remainder  =  -125 + 75 + 25 + 2

Remainder  =  -23

Example 4 :

Using Factor theorem, check whether (x-2) is a factor of the polynomial x⁴ - 3x³ + 2x² + 8x - 16.

Solution :

Step 1 :

Let P(x)  =  x⁴ - 3x³ + 2x² + 8x - 16

If x - 2 = 0, then x  =  2.

Step 2 :

Using Factor theorem, to check whether (x-2) is factor of P(x), plug x = 2 in P(x).

P(2)  =  (2)⁴ - 3(2)³ + 2(2)² + 8(2) - 16

P(2)  =  16 - 24 + 8 + 16 - 16

P(2)  =  0

Because P(2) = 0, by Factor theorem, (x-2) is a factor of the polynomial P(x).

Example 5 :

Using Factor theorem, check whether (x+4) is a factor of the polynomial x² - 8x + 16.

Solution :

Step 1 :

Let P(x)  =  x² - 8x + 16

If x + 4 = 0, then x  =  -4.

Step 2 :

Using Factor theorem, to check whether (x+4) is factor of P(x), plug x = -4 in P(x).

P(-4)  =  (-4)² + 8(-4) + 16

P(-4)  =  16 - 32 + 16

P(-4)  =  0

Because P(-4) = 0, by Factor theorem, (x+4) is a factor of the polynomial P(x).

Example 6 :

Using Factor theorem, check whether (x-3) is a factor of the polynomial x³ - 2x² + 5x + 6.

Solution :

Step 1 :

Let P(x)  =  x³ - 2x² + 5x + 6

If x - 3 = 0, then x  =  3.

Step 2 :

Using Factor theorem, to check whether (x-3) is factor of P(x), plug x = 3 in P(x).

P(3)  =  (3)³ - 2(3)² + 5(3) + 6

P(3)  =  27 - 2(9) + 15 + 6

P(3)  =  27 - 18 + 15 + 6

P(3)  =  30 ≠ 0

Because P(3) ≠ 0, by Factor theorem, (x-3) is not a factor of the polynomial P(x).

After having gone through the stuff given above, we hope that the students would have understood "Remainder and Factor theorem".

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