In this section , we shall study a simple and an elegant method of finding the remainder.

In the case of divisibility of a polynomial by a linear polynomial we use a well known theorem called Remainder Theorem.

**Remainder Theorem : **

If a polynomial p(x) of degree greater than or equal to one is divided by a linear polynomial (x–a) then the remainder is p(a), where a is any real number.

**Significance of Remainder theorem :**

It enables us to find the remainder without actually following the cumbersome process of long division.

**Note : **

(i) If p(x) is divided by (x+a), then the remainder is

p(– a)

(ii) If p(x) is divided by (ax–b), then the remainder is

p(b/a)

(iii) If p(x) is divided by (ax+b), then the remainder is

p(-b/a)

**Example 1 :**

Using Remainder Theorem, find the remainder when

f(x) = x^{3} + 3x^{2} + 3x + 1

is divided by (x + 1).

**Solution : **

Here, the divisor is (x + 1).

Equate the divisor to zero.

x + 1 = 0

Solve for x.

x = -1

To find the remainder, substitute -1 for x into the function f(x).

f(-1) = (-1)^{3} + 3(-1)^{2} + 3(-1) + 1

f(-1) = -1 + 3(1) - 3 + 1

f(-1) = -1 + 3 - 3 + 1

f(-1) = 0

So, the remainder is 0.

**Example 2 :**

Using Remainder Theorem, find the remainder when

f(x) = x^{3} - 3x + 1

is divided by (2 - 3x).

**Solution : **

Here, the divisor is (2 - 3x).

Equate the divisor to zero.

2 - 3x = 0

Solve for x.

-3x = -2

x = 2/3

To find the remainder, substitute 2/3 for x into the function f(x).

f(2/3) = (2/3)^{3} - 3(2/3) + 1

f(2/3) = 8/27 - 2 + 1

f(2/3) = 8/27 - 1

f(2/3) = 8/27 - 27/27

f(2/3) = (8 - 27)/27

f(2/3) = -19/27

So, the remainder is -19/27.

**Example 3 :**

For what value of k is the polynomial

2x^{4} + 3x^{3} + 2kx^{2} + 3x + 6

is divisible by (x + 2).

**Solution : **

Let

f(x) = 2x^{4} + 3x^{3} + 2kx^{2} + 3x + 6

Here, the divisor is (x + 2).

Equate the divisor to zero.

x + 2 = 0

Solve for x.

x = -2

To find the remainder, substitute -2 for x into the function f(x).

f(-2) = 2(-2)^{4} + 3(-2)^{3} + 2k(-2)^{2} + 3(-2) + 6

f(-2) = 2(16) + 3(-8) + 2k(4) - 6 + 6

f(-2) = 32 - 24 + 8k - 6 + 6

f(-2) = 8 + 8k

So, the remainder is (8 + 8k).

If f(x) is exactly divisible by (x + 2), then the remainder must be zero.

Then,

8 + 8k = 0

Solve for k.

8k = -8

k = -1

Therefore, f(x) is exactly divisible by (x+2) when k = –1.

If p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number then

(i) p(a) = 0 implies (x - a) is a factor of p(x).

(ii) (x - a) is a factor of p(x) implies p(a) = 0.

**Note : **

(i) (x - a) is a factor of p(x), if p(a) = 0.

(ii) (x + a) is a factor of p(x), if p(-a) = 0.

(iii) (ax + b) is a factor of p(x), if p(-b/a) = 0.

(iv) (x - a)(x - b) is a factor of p(x), if

p(a) = 0 and p(b) = 0

**Example 1 :**

Using Factor Theorem, show that (x + 2) is a factor of

x^{3} - 4x^{2} - 2x + 20

**Solution : **

Let

f(x) = x^{3} - 4x^{2} - 2x + 20

Equate the factor (x + 2) to zero.

x + 2 = 0

Solve for x.

x = -2

By Factor Theorem,

(x + 2) is factor of f(x), if f(-2) = 0

Then,

f(-2) = (-2)^{3} - 4(-2)^{2} - 2(-2) + 20

f(-2) = -8 - 4(4) + 4 + 20

f(-2) = -8 - 16 + 4 + 20

f(-2) = 0

Therefore, (x + 2) is a factor of x^{3} - 4x^{2} - 2x + 20.

**Example 2 :**

Is (3x - 2) a factor of 3x^{3} + x^{2} - 20x + 12 ?

**Solution : **

Let

f(x) = 3x^{3} + x^{2} - 20x + 12

Equate the factor (3x + 2) to zero.

3x - 2 = 0

Solve for x.

3x = 2

x = 2/3

By Factor Theorem,

(3x - 2) is factor of f(x), if f(2/3) = 0

Then,

f(2/3) = 3(2/3)^{3} + (2/3)^{2} - 20(2/3) + 12

f(2/3) = 3(8/27) + 4/9 - 40/3 + 12

f(2/3) = 8/9 + 4/9 - 40/3 + 12

f(2/3) = 8/9 + 4/9 - 120/9 + 108/9

f(2/3) = (8 + 4 - 120 + 108) / 9

f(2/3) = (120 - 120) / 9

f(2/3) = 0

Therefore, (3x - 2) is a factor of 3x^{3} + x^{2} - 20x + 12.

**Example 3 :**

Find the value of m, if (x - 2) is a factor of the polynomial

2x^{3} - 6x^{2} + mx + 4

**Solution : **

Let

f(x) = 2x^{3} - 6x^{2} + mx + 4

Equate the factor (x - 2) to zero.

x - 2 = 0

Solve for x.

x = 2

By Factor Theorem,

(x - 2) is factor of f(x), if f(2) = 0

Then,

f(2) = 0

2(2)^{3} - 6(2)^{2} + m(2) + 4 = 0

f(2) = 2(8) - 6(4) + 2m + 4 = 0

f(2) = 16 - 24 + 2m + 4 = 0

f(2) = 2m - 4 = 0

2m = 4

m = 2

Therefore (x - 2) is a factor of f(x), when m = 2.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**