Using the algebraic expressions, their addition, subtraction, multiplication of the polynomial of a squares. How to factorize some algebraic expressions. You may recall the algebraic identities using the polynomial of a square

(x + y)^{2} = x^{2} + 2xy + y^{2}

(x – y)^{2} = x^{2} – 2xy + y^{2}

and x^{2} – y^{2} = (x + y) (x – y)

and their use in factorization. we shall start our study with a particular type of algebraic expression, called polynomial, of square and the terminology related to it.

Polynomial of a square in one variable and their degrees.If *p*(*x*) is a polynomial in *x*, The power of p(x) is called **the degree of** **the polynomial **p(x). For example, 4*x *+ 2 is a polynomial in the variable *x *of degree 1, 2y^{2} – 3y + 4 is a polynomial in the variable *y *of degree 2, 5x^{3} – 4x^{2} + x – 2 is a polynomial in the variable *x *of degree 3 and 7u^{6} – `3/2` u^{4}+u-8 is a polynomial

in the variable u of degree 6. expression like `1/(x-1)` ,`1/(x2+2x+3) ` etc. Are not a polynomial.

## Addition of polynomial of a square

**Addition coefficient polynomial of a square:
**

We add two polynomials by adding the coefficient of like powers.

**Example 1:**

Find the sum of 2x^{4 }– 3x^{2 }+ 5x + 3 and 4x + 6x^{3 }– 6x^{2 }– 1.

**Solution:**

The associative and distributive properties of real numbers.

(2x^{4 }– 3x^{2 }+ 5x + 3) + (6x^{3 }– 6x^{2 }+ 4x – 1) = 2x^{4 }+ 6x^{3 }– 3x^{2 }– 6x^{2 }+ 5x + 4x + 3 – 1

= 2x^{4 }+ 6x^{3 }– (3+6)x^{2 }+ (5+4)x + 2

= 2x^{4 }+ 6x^{3 }– 9x^{2 }+ 9x + 2.

** subtraction coefficient polynomial of a square:**

We subtract polynomials like addition of polynomials.

**Example 2:**

Subtract 2x^{3 }– 3x^{2 }– 1 from x^{3 }+ 5x^{2 }– 4x – 6.

**Solution:**

Using the properties of associate and distributive.

(x^{3 }+ 5x^{2 }– 4x – 6) – (2x^{3 }– 3x^{2 }– 1) = x^{3 }+ 5x^{2 }– 4x – 6 – 2x^{3 }+ 3x^{2 }+ 1

= x^{3 }– 2x^{3 }+ 5x^{2 }+ 3x^{2 }– 4x – 6 + 1

= (x^{3 }– 2x^{3}) + (5x^{2 }+ 3x^{2}) + (–4x) + (–6+1)

= –x^{3 }+ 8x^{2 }– 4x – 5.

## multipliying polynomial of a square

To find the multiplication of two polynomials, we use the distributive properties and the law of exponents.

**Example 3:**

Find the product of x^{3 }– 2x^{2 }– 4 and 2x^{2 }+ 3x – 1 .

**Solution:**

(x^{3 }– 2x^{2 }– 4) (2x^{2 }+ 3x – 1)

= x^{3 }(2x^{2 }+ 3x – 1) + (–2x^{2}) (2x^{2 }+ 3x – 1) + (–4) (2x^{2 }+ 3x – 1)

= (2x^{5 }+ 3x^{4 }– x^{3}) + (–4x^{4 }– 6x^{3 }+ 2x^{2}) + (–8x^{2 }– 12x + 4)

= 2x^{5 }+ 3x^{4 }– x^{3 }– 4x^{4 }– 6x^{3 }+ 2x^{2 }– 8x^{2 }– 12x + 4

= 2x^{5 }+ (3x^{4 }– 4x^{4}) + (–x^{3 }– 6x^{3}) + (2x^{2 }– 8x^{2}) + (–12x) +4

= 2x^{5 }– x^{4 }– 7x^{3 }– 6x^{2 }– 12x + 4.