# Factorization of Polynomials Using Algebraic Identities

#### Summary

**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}= a^{2}+ 2ab + b^{2} - (a – b)
^{2}= a^{2}– 2ab + b^{2} - a
^{2}– b^{2}= (a + b)(a – b) - (x + a)(x + b) = x
^{2}+ (a + b)x + ab - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 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}= a^{3}+ b^{3}+ 3ab (a + b) - (a – b)
^{3}= a^{3}– b^{3}– 3ab (a – b) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2}) - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}+ c^{3 }– 3abc = (a + b + c)(a^{2}+ b^{2}+ c^{2}– ab – bc – ca)

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#### Questions & Answers*arrow_upward*

### 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