Math's: Special Products: Factorization of Polynomials, HCF and LCM of Polynomials

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Factorization of Polynomials

  • Factorization of a polynomial is a process of writing the polynomial as a product of two (or more) polynomials. Each polynomial in the product is called a factor of the given polynomial.

  • A polynomial will be said to be completely factorized if none of its factors can be further expressed as a product of two polynomials of lower degree and if the integer coefficients have no common factor other than 1 or –1. For Instance, the complete factorization of is On the other hand the factorization of is not complete since the factor can be further factorised as . Thus, complete factorization of is .

Factorization by Distributive Property

Here, the polynomial is factorized by finding and taking out the highest common factor(s) from the terms and distributing it with those terms.

Distributive Property

Distributive Property

Ex- factorize

Sol-

Factorization Involving Difference of Two Squares

We know that . Therefore and are factors of

Ex- factorize

Sol-

Factorization of a Perfect Square Trinomial

When the terms of a polynomial are either in the form of or , the two factors are identical, each being and

In each of the two respective cases.

Ex- factorize

Sol.

As the given polynomial is in the form of , the two factors of

will be each.

That is

Factorization of a Perfect Cube Polynomial

Polynomials in the form of or are factorised as the perfect cubes of and respectively.

Ex- Factorize-

Sol-

Factorization of Polynomial Involving Sum or Difference of Two Cubes

Polynomials in the form of or are factorized as

and respectively.

Ex- Factorize

Sol-

Factorization of Trinomial by Splitting the Middle Term

Expressions in the form of can be factorized by multiplying the coefficient of in the first term with the last term and finding two such factors of this product that their sum is equal to the coefficient of in the second (middle) term. In other words, we are to determine two such factors of ac so that their sum is equal to b.

Ex- factories

Sol-

LCM and HCF of Polynomials

  • The Highest Common Factor (HCF) of two or more given polynomials is the product of the polynomial(s) of highest degree and greatest numerical coefficient each of which is a factor of each of the given polynomials. For example, the HCF of and is

  • The Lowest Common Multiple (LCM) of two or more polynomials is the product of the polynomial(s) of the lowest degree and the smallest numerical coefficient which are multiples of the corresponding elements of each of the given polynomials. For example, the HCF of and is .

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