# Factorization of Polynomials Using Algebraic Identities

#### Summary

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Factorisation

If g(x) and h(x) are two polynomials whose product is p(x). This can be written as p(x) = g(x) . h(x). g(x) and h(x) are called the factors of the polynomial p(x).

The process of resolving a given polynomial into factors is called factorisation. A non-zero constant is a factor of every polynomial.

Algebraic Identities

Polynomials can be factorised using algebraic identities.
A polynomial of degree two is called a quadratic polynomial. The identities used to factorise the quadratic polynomials are:

• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• a2 – b2 = (a + b)(a – b)
• (x + a)(x + b) = x2 + (a + b)x + ab
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

A polynomial of degree three is called a cubic polynomial. The algebraic identities used in factorising a cubic polynomial are:

• (a + b)3 = a3 + b3 + 3ab (a + b)
• (a – b)3 = a3 – b3 – 3ab (a – b)
• a3 + b3 = (a + b)(a2 – ab + b2)
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 + c– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

### 1 . If x + 1/x = 3 , then find the value of x6+1/x6.

give the question properly

### 2 . x3 +1/x3 = 110,then x+1/x =

Given x 3+ 1/x 3 =110
Recall that (x+1/x) 3 = x 3+ 1/x 3 + 3(x+1/x)...

### 3 . If x + y + z = 8 and xy + yz + zx = 20 find the value of x3 + y3 + z3 - 3xyz.

Given x + y + z = 8 and xy + yz + zx = 20
Consider, x + y + z = 8
Squaring on both sides, we get
(x + y + z) 2...

### 4 . If a2+b2+c2-ab-bc-ca=0,prove that a=b=c.

Consider, a 2 + b 2 + c 2 ? ab ? bc ? ca = 0
Multiply both sides with 2, we get
2( a...

### 5 . If x4+1/x4 =119 then find x3-1/x3

Now subtract 2 from both the sides, we get

Cubing on both the sides we get