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Square of a Binomial

A polynomial is a function having usually many terms in one or even more variables.
On the basis of number of terms, the polynomial may be classified as:

1) Monomial (polynomial with one term)
2) Binomial (polynomial having two terms)
3) Trinomial (polynomial containing three terms)
More than this is already known as a polynomial.
In this article, we going to focus basically on squares of binomials and trinomials. The general form of square of binomials can be written as (x + y)$^{2}$. Similarly, the general form of square of trinomials is to be written as (x + y + z)$^{2}$. Where, x, y and z are variable which may or may not be associated with constant coefficients.

The formulae for finding square of a binomial are given below:
$(x + y)^{2} = x^{2} + y^{2} + 2 xy$
$(x - y)^{2} = x^{2} + y^{2} - 2 xy$
These fomulae may be be derived as:

(x + y)^${2}$

= (x + y) (x + y)

= x.x  + x.y + y.x + y.y

= $x^{2} + y^{2} + 2 xy$

(x - y)^${2}$

= (x - y) (x - y)

= x.x  + x.(-y) + (-y).x + (-y).(-y)

= x.x  - x.y - y.x + y.y

= $x^{2} + y^{2} - 2 xy$

In order to find square of a binomial, we assume first term as x and second term as y. Then apply any of the above suitable formula. If any term is connected with constant coefficients, then that coefficient should be considered as a part of that term and dealt accordingly.

Understand the following examples carefully.

1) $(a + 5)^{2}$

Apply formula for $(x + y)^{2}$

= $a^{2} + 5^{2} + 2 \times a \times 5$

= $a^{2} + 25 + 10 a$

2) $(3x - 4)^{2}$

Apply formula for $(x - y)^{2}$

= $(3x)^{2} + 4^{2} - 2 \times 3x \times 4$

= $9x^{2} + 16 - 24x$

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Perfect Square Trinomials

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A perfect square trinomial is a function in one or more variables which contains three (tri means three) terms. Two terms in a perfect square trinomial should be a perfect squares of some number or variable with constant coefficient. The third term has to be the twice of the product of these perfect squares. Have a look at the following diagram illustrating a perfect square trinomial:

Perfect Square Trinomial

For Example:

x$^{2}$ + 12x + 36

= x$^{2}$ + 12x + 6$^{2}$

= x$^{2}$ + 2. x. 6 + 6$^{2}$

This is a perfect square trinomial.

Now, this can be written in the form of (x + y)$^{2}$.

Completing The Square

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"Completing the square" is a quite common method of factorization. In this method, one has to recognize and form the perfect square trinomial and then substitute back the formula of square of the binomial. This method involves the step mentioned below:

Step 1: We have to make sure about the coefficient of highest power in the polynomial is one.

For Example: a$^{2}$ + 8a - 4 = 0

Here, a has the coefficient 1.

If it is not 1, we need to divide every term by the leading coefficient making the coefficient of highest power as one.

Step 2: Write the middle term as the product of 2 and the perfect square term, i.e.
a$^{2}$ + 2. a . 4 - 4 = 0

Step 3: Now, the third number should be the perfect square of the remaining number in second term. In above example, the remaining number after 2a is 4, so third term should be the square of 4, i.e. 16.

In order to make third term a square term, we need to add and subtract a number in the equation so as to make it square of required term. i.e. in above example, we should add and subtract 
4$^{2}$ = 16.

a$^{2}$ + 2. a . 4 - 4 = 0

a$^{2}$ + 2. a . 4 + 16 - 16- 4 = 0

Step 4: Separate the perfect square trinomial in this way.

(a$^{2}$ + 2. a . 4 + 16) - 20 = 0

Step 5: Apply the suitable formula of square of the binomial.

(a$^{2}$ + 2. a . 4 + 16) - 20 = 0

(a + 4)$^{2}$ - 20 = 0

Step 6: Now, factorize the equation thus obtained.

(a + 4)$^{2}$ - 20 = 0

$(a + 4)^{2} - (\sqrt{20})^{2} = 0$

$(a + 4 + 2\sqrt{5}) (a + 4 - 2\sqrt{5}) = 0$

$a + 4 + 2\sqrt{5} = 0$ and $a + 4 - 2\sqrt{5} = 0$

a = - 4 - $2\sqrt{5}$

a = - 4 + $2\sqrt{5}$

Square of a Trinomial

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A trinomial is an expression having three terms. The formula for the square of a trianomial is given below:
$(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx$For Example:

(a - 2b + c)$^{2}$

=  a$^{2}$ + (-2b)$^{2}$ + c$^{2}$ + 2 a (-2b) + 2 (-2b) c + 2 c a

=  a$^{2}$ + 4b$^{2}$ + c$^{2}$ - 4ab - 4bc + 2ca

Examples

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Have a look at the example based on this article:

Example 1: Evaluate (6m - 7n)$^{2}$

Solution: Using the formula $(x - y)^{2} = x^{2} + y^{2} - 2 xy$

$(6m - 7n)^{2}$

= $(6m)^{2} + (7n)^{2} - 2 . 6m . 7n$

= $36 m^{2} + 49 n^{2} - 84 m n$

Example 2: Factorize $(a + b)^{2} - 12 (a + b) + 36$

Solution : $(a + b)^{2} - 12 (a + b) + 36$

= $(a + b)^{2} - 2 (a + b) . 6 + 6^{2}$

This is the form of perfect square trinomial of $x^{2} + y^{2} - 2 xy$ which is equal to $(x - y)^{2}$, hence

= $[(a + b) - 6]^{2}$

= $(a + b - 6)^{2}$

Example 3: Solve $(5x - y - 2z)^{2}$

Solution: Using the formula $(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx$, we get

$(5x - y - 2z)^{2}$

= (5x)$^{2}$ + (-y)$^{2}$ + (-2z)$^{2}$ + 2 (5x) (-y) + 2 (-y) (-2z) + 2 (-2z) (5x)

= 25 x$^{2}$ + y$^{2}$ + 4 z$^{2}$ - 10 xy + 4 yz - 20 xz
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