In this section, you will learn how to find square root of a polynomial using long division.

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**Note : **

Before proceeding to find the square root of a polynomial, one has to ensure that the degrees of the variables are in descending or ascending order.

**Example 1 : **

Find the square root of the following polynomial :

x^{4} - 4x^{3} + 10x^{2} - 12x + 9

**Solution :**

Therefore the square root of the given polynomial is

|x^{2} - 2x + 3|

**Example 2 :**

Find the square root of the following polynomial :

4x^{4} + 8x^{3} + 8x^{2} + 4x + 1

**Solution :**

Therefore the square root of the given polynomial is

|2x^{2} + 2x + 1|

**Example 3 :**

Find the square root of the following polynomial :

9x^{4} - 6x^{3} + 7x^{2} - 2x + 1

**Solution :**

Therefore the square root of the given polynomial is

|3x^{2} - x + 1|

**Example 4 :**

Find the square root of the following polynomial :

4 + 25x^{2} - 12x - 24x^{3} + 16x^{4}

**Solution :**

First arrange the term of the polynomial from highest exponent to lowest exponent and find the square root.

Then,

**16x ^{4} - 24x^{3} + 25x^{2} - 12x + 4**

Therefore the square root of the given polynomial is

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

**Example 5 :**

Find the values of a and b if the following polynomial is a perfect square

**4x ^{4} - 12x^{3} + 37x^{2} + ax + b**

**Solution :**

Because the given polynomial is a perfect square,

a + 42 = 0 and b - 49 = 0

Solving the above equations for a and b, we get

a = -42

b = 49

**Example 6 :**

Find the values of a and b if the following polynomial is a perfect square

**x ^{4} - 4x^{3} + 10x^{2} - ax + b**

**Solution :**

Because the given polynomial is a perfect square,

-a + 12 = 0 and b - 9 = 0

Solving the above equations for a and b, we get

a = 12

b = 9

**Example 7 :**

Find the values of a and b if the following polynomial is a perfect square

**ax ^{4} + bx^{3} + 109x^{2} - 60x + 36**

**Solution :**

Here a and b are being the coefficients of x^{4} and x^{3} respectively.

To solve for a and b, always they have to come at last.

So, write the given polynomial from lowest exponent to highest exponent.

36 - 60x + 109x^{2} + bx^{3} + ax^{4}

Because the given polynomial is a perfect square,

bx^{3} + 70x^{3} = 0 and ax^{4} - 49x^{4} = 0

Solve the above equations for a and b.

bx (b + 70)x Divide each side by x b + 70 = 0 b = -70 |
ax (a - 49)x Divide each side by x a - 49 = 0 a = 49 |

Therefore,

a = 49

b = -70

**Example 8 :**

Find the values of a and b if the following polynomial is a perfect square

**ax ^{4} - bx^{3} + 40x^{2} + 24x + 36**

**Solution :**

Here a and b are being the coefficients of x^{4} and x^{3} respectively.

To solve for a and b, always they have to come at last.

So, write the given polynomial from lowest exponent to highest exponent.

36 + 24x + 40x^{2} - bx^{3} + ax^{4}

Because the given polynomial is a perfect square,

-bx^{3} - 12x^{3} = 0 and ax^{4} - 9x^{4} = 0

Solve the above equations for a and b.

-bx (b + 12)(-x Divide each side by (-x b + 12 = 0 b = -12 |
ax (a - 9)x Divide each side by x a - 9 = 0 a = 9 |

Therefore,

a = 9

b = -12

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