**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.

**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 7x⁴ - x²- 3x + 9 is divided by (x-6).

**Solution : **

**Step 1 :**

**Let P(x) = **7x⁴ - 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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