**Finding the Square Root of a Polynomial by Long Division Method**

The long division method in finding the square root of a polynomial is useful when the degree of the polynomial is higher.

Here we are going to see how to find square root of a polynomial of degree 4.

Find the square root of the following polynomials by division method

(i) x^{4} −12x^{3} + 42x^{2} −36x + 9

**Step 1 :**

x^{4} has been decomposed into two equal parts x^{2} and x^{2}.

**Step 2 :**

Multiplying the quotient (x^{2}) by 2, so we get 2x^{2}. Now bring down the next two terms -12x^{3} and 42x^{2}.

By dividing -12x^{3 }by 2x^{2}, we get -6x. By continuting in this way, we get the following steps.

Hence the square root of x^{4} −12x^{3} + 42x^{2} −36x + 9 is x^{2} - 6x + 3

(ii) 37x^{2} −28x^{3} + 4x^{4} + 42x + 9

**Solution :**

First let us arrange the given polynomial from greatest order to least order.

4x^{4} −28x^{3 }+ 37x^{2 }+ 42x + 9

Hence the square root of 37x^{2} −28x^{3} + 4x^{4} + 42x + 9 is 2x^{2} - 7x - 3.

(iii) 16x^{4} + 8x^{2} + 1

**Solution :**

Hence the square root of 37x^{2} −28x^{3} + 4x^{4} + 42x + 9 is 4x^{2} + 0x + 1.

(iv) 121x^{4} − 198x^{3} − 183x^{2} + 216x + 144

**Solution :**

Hence the square root of 121x^{4} − 198x^{3} − 183x^{2} + 216x + 144 is 11x^{2} + 9x + 12.

**Question 2 :**

Find the square root of the expression

(x^{2}/y^{2}) - 10x/y + 27 - (10y/x) + (y^{2}/x^{2})

**Solution :**

By taking L.C.M, we get

(x^{4} - 10x^{3}y + 27x^{2}y^{2} - 10xy^{3}+ y^{4})/x^{2}y^{2}

= √(x^{4} - 10x^{3}y + 27x^{2}y^{2} - 10xy^{3}+ y^{4})/√x^{2}y^{2}

= (x^{2} - 5xy + y^{2})/xy

= (x/y) - 5 + (y/x)

Hence the square root of the polynomial (x^{2}/y^{2}) - 10x/y + 27 - (10y/x) + (y^{2}/x^{2}) is (x/y) - 5 + (y/x).

Let us look into the next example on "Finding the Square Root of a Polynomial by Long Division Method".

**Question 1 :**

Find the values of a and b if the following polynomials are perfect squares

(i) 4x^{4} −12x^{3} + 37x^{2} + bx + a

**Solution :**

By equating the coefficients of x, we get

b = -42

By equating the constant terms, we get

a = 49

Hence the values of a and b are -49 and 42 respectively.

(ii) ax^{4} + bx^{3} + 361x^{2} + 220x + 100

**Solution :**

Equating the coefficients of x^{3}, we get

b = 264

By equating the coefficients of x^{4}, we get

a = 144

Hence the values of a and b are 144 and 264 respectively.

**Question 2 :**

Find the values of m and n if the following expressions are perfect sqaures

(i) (1/x^{4}) - (6/x^{3}) + (13/x^{2}) + (m/x) + n

**Solution :**

By taking L.C.M, we get

(1 - 6x + 13x^{2} + mx^{3} + nx^{4})/x^{4}

By equating the coefficients of x^{3}, we get

m = -12

By equating the coefficients of x^{4}, we get

n = 4

Hence the values of m and n are 6 and 4 respectively.

(ii) x^{4} − 8x^{3} + mx^{2} + nx + 16

**Solution :**

By equating the constant term, we get

[(m - 16)/2]^{2} = 16

(m - 16)/2 = 4

m - 16 = 8

m = 8 + 16 = 24

By equating the coefficients of x, we get

n = -4(m - 16)

n = -4(24 - 16)

n = -4(8) = -32

Hence the values of m and n are 24 and -32.

After having gone through the stuff given above, we hope that the students would have understood, "Finding the Square Root of a Polynomial by Long Division Method".

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