• Factorization is expressing any algebraic equation as product of its factors.
  • These factors can be numbers, variables or algebraic expressions.
  • An irreducible factor is a factor which cannot be expressed further as a product of factors.
  • Common factor method of factorization:
    1. Write each term of the expression as a product of irreducible factors.
    2. Separate the common factor terms.
    3. Combine the remaining factors in each term in accordance with the distributive law.
  • Terms must be grouped in a way that each group of terms have common factors. This is the method of regrouping.
  • We need to observe expression and identify the desired grouping by trail and error.
  • Common identities for factorization:
    1. (a + b)2 = a2 + 2ab + b2
    2. (a – b)2 = a2 – 2ab + b2
    3. (a + b) (a – b) = a2 - b2
    4. (y + a) (y + b) = y2 + (a + b)y + ab
  • In expressions which have factors of the type (y + a) (y + b), remember the numerical term gives ab. Factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of y.
  • Division of numbers is inverse of its multiplication.
  • In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
  • In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.
  • In the case of divisions of algebraic expressions that we studied in this chapter, we have:

Dividend = Divisor × Quotient.

In general, however, the relation is:

Dividend = Divisor × Quotient + Remainder

Thus, we have considered in the present chapter only those divisions in which the remainder is zero.

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