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FACTORISATION data

Facterization

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FACTORISATION data

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  1. FACTORISATION Defination:The process on writing analgebric expression as a product of two or more algebraic expression whic are called factors of first algebraic expression is called Factorisation. P r e p a r e d b y T C H A I T A N Y A

  2. FACTORISATION What is Factorisation in Mathematics? What is Factorisation in Mathematics? Factorisation of an algebraic expression means writing the given expression as a product of its factors. These factors can be numbers, variables, or an algebraic expression. To the factor, a number means to break it up into numbers that can be multiplied to get the original number. For example, 24 = 4 × 6 4 and 6 are the factors of 24 9 = 3 × 3 3 is the factor of 9 Also, numbers can be factorized into different combinations. There are different ways to find the Factors of a Number. To find the factors of an integer is an easy method but to find the factors of algebraic equations is not that easy. So let us learn to find the factors of quadratic polynomial Prepared by T CHAITANYA

  3. FACTORISATION  FACTORISATION IN ALGEBRA: The numbers 1, 2, 6, and 12 are all factors of 12 because they divide 12 without a remainder. It is an important process in algebra which is used to simplify expressions, simplify fractions, and solve equations. It is also called as Algebra factorization. Terms and Factors What is a Term? It is something which is to be added or subtracted (subtracting is adding a negative number) in an expression. If 2x + 7 is an expression, then 2x and 7 are the terms. Sum = term + term Prepared by T CHAITANYA

  4. FACTORISATION  TERMS: What is a TERM? It is something which is to be added or subtracted (subtracting is adding a negative number) in an expression. If 2x + 7 is an expression, then 2x and 7 are the terms Prepared by T CHAITANYA

  5. FACTORISATION  TERMS: Small text It is something that is to be multiplied in an expression. Product = factor × factor For example : p = 4(2q – 6) 4 and 2q – 6 are the factors whereas 2q and 6 are the terms Prepared by T CHAITANYA

  6. FACTORISATION  Basic Factorisation Formula: a2 – b2 = (a – b)(a + b) (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 a3 – b3 = (a – b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 – ab + b2) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca Prepared by T CHAITANYA

  7. FACTORISATION  Basic Factorisation Formula: There are four methods to factorise the algebraic expressions. Common factors method Regrouping terms method Factorisation using identities Factors of the form (x+a) (x+b) Prepared by T CHAITANYA

  8. FACTORISATION  Methods of Common Factors: Method of Common Factors In this method, we simply take out the common factors among each term of the given expression. Example: Factorise 3x + 9. Since, 3 is the common factor for both the terms 3x and 9, thus taking 3 as a common factor we get; 3x + 9 = 3(x+3). Prepared by T CHAITANYA

  9. FACTORISATION  Regrouping of Terms Method: Regrouping means rearranging the given expression based on the like terms or similar terms. For example, 2xy + 3x + 2y + 3 can be rearranged as: 2xy + 3x + 2y + 3 Expanding the terms into factor form. = 2 × x × y + 3 × x + 2 × y + 3 Rearrange to get the common factor = x × (2y + 3) + 1 × (2y + 3) Now (2y + 3) is the common factor we can take out. = (2y + 3) (x + 1) Thus, these are the required factors. P r e p a r e d b y T C H A I T A N Y A

  10. FACTORISATION  Factorisation using identities: By using the common identities, we can factorise the given expression. Example: Factorise 4x2 – 9. Solution: By using the algebraic identities, we know; a2 – b2 = (a – b) (a+b) Hence, we can write, 4x2 – 9 = (2x)2 – 32 = (2x + 3) (2x – 3) Prepared by T CHAITANYA

  11. FACTORISATION  Factorization Formula for a Quadratic Polynomial: A “quadratic” is a polynomial that is written like “ax2 + bx + c”, where “a”, “b”, and “c” are just numbers. For an easy case of factoring, you can identify the two numbers that will not only multiply to equal the constant term “c” but also add up to equal “b,” the coefficient on the x-term. Factorising formulas algebra is especially important when solving quadratic polynomial When reducing formulas we normally have to remove all the brackets, but in particular cases, for example with fractional formulas, sometimes we can use factorisation to shorten a formula. P r e p a r e d b y T C H A I T A N Y A

  12. FACTORISATION  Types of Factoring polynomials: There are six different methods to factorising polynomials. The six methods are as follows: Greatest Common Factor (GCF) Grouping Method Sum or difference in two cubes Difference in two squares method General trinomials Trinomial method Prepared by T CHAITANYA

  13. FACTORISATION  How to Solve Polynomials? There are a certain number of methods by which we can solve polynomials. Let us discuss these methods. Prepared by T CHAITANYA

  14. FACTORISATION  Greatest Common Factor: We have to find out the greatest common factor, of the given polynomial to factorise it. This process is nothing but a type of reverse procedure of distributive law, such as; p( q + r) = pq + pr But in the case of factorisation, it is just an inverse process; pq + pr = p(q + r) where p is the greatest common factor. Prepared by T CHAITANYA

  15. FACTORISATION  Factoring Polynomials By Grouping This method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros. Prepared by T CHAITANYA

  16. FACTORISATION  Factoring Using Identities: The factorisation can be done also by using algebraic identities. The most common identities used in terms of the factorisation are: (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 a2 – b2= (a + b)(a – b) Prepared by T CHAITANYA

  17. FACTORISATION  Factor theorem: For a polynomial p(x) of degree greater than or equal to one, x-a is a factor of p(x), if p(a) = 0 If p(a) = 0, then x-a is a factor of p(x) Where ‘a’ is a real number. P r e p a r e d b y T C H A I T A N Y A

  18. FACTORISATION  Factoring Polynomial with Four Terms: Let us learn how to factorize the polynomial having four terms. For example, x3 + x2 – x – 1 is the polynomial. Break the given polynomial into two parts first. (x3 + x2)+( –x – 1) Now find the highest common factor from both the parts and take that factor out of the bracket. We can see, from the first part, x2 is the greatest common factor and from the second part we can take out the minus sign. Prepared by T CHAITANYA

  19. FACTORISATION  How to factor a polynomial?: A polynomial can be factorised using different methods such as finding the greatest common factor of all the terms, splitting the polynomial into two parts, using algebraic identities, etc. Prepared by T CHAITANYA

  20. FACTORISATION  How to factorise the given algebraic expression? There are a few methods or tricks to remember. Reduce the expression by the greatest common factor. Check if there are perfect squares or differences of squares. Use the algebraic formulas or identities to factorise the expression. Prepared by T CHAITANYA

  21. FACTORISATION  How to factorise quadratic equation? How to factorise quadratic equation? The most common method to solve quadratic equations is complete the square method or directly use the quadratic formula. P r e p a r e d b y T C H A I T A N Y A

  22. FACTORISATION  What is the example of factorisation? Suppose, 3x+6xy+30+60y is an expression. Taking the common factors we get; 3x(1+2y)+ 30 (1+2y) (1+2y) (3x+30) 3(1+2y) (x+10) Thus, the required factors are: 3(1+2y)(x+10) Prepared by T CHAITANYA

  23. FACTORISATION  Factorization using common factors: To factorize an algebraic expression, the highest common factors of the terms of the given algebraic expression are determined and then we group the terms accordingly. In simple terms, the reverse process of expansion of an algebraic expression is its factorization. P r e p a r e d b y T C H A I T A N Y A

  24. FACTORISATION  Factorization by regrouping terms In some algebraic expressions, not every term may have a common factor. For instance, consider the algebraic expression 12a + n -na – 12. The terms of this expression do not have a particular factor in common but the first and last term has a common factor of ‘12’ similarly second and third term has n as a common factor. Prepared by T CHAITANYA

  25. FACTORISATION  Factorizing Expressions using standard identities An equality relation which holds true for all the values of variables in mathematics is known as an identity. Consider the following identities: On substituting any value of a and b, both sides of the given equations remain the same. Therefore, these equations are called identities. Prepared by T CHAITANYA

  26. FACTORISATION  What is the example of factorisation? What is the rule for factorisation? If there are four terms in a given expression, then take out the common terms. Suppose, xy-5y+10-2x is given to us. So we can write, y(x-5) + 2(5-x) Or y(x-5)-2(x-5) Now writing them as product of factors, we get; (x-5) (y-2) Prepared by T CHAITANYA

  27. FACTORISATION  Solved Examples Here are some maths factorisation example questions and how to factorise the quadratic polynomial are explained in detail. Q.1: Factorise the Quadratic Polynomial: x2 + 7x + 6 The constant term is 6, which can be written as the product of 2 and 3 or of 1 and 6. But 2 + 3 = 5, so 2 and 3 are not the numbers I need in this case. On the other hand, 1 + 6 = 7, so you can use 1 and 6: x2 + 7x + 6 = (x+1)(x+6) Note that the order doesn’t matter in multiplication, so the above answer can be written as “(x + 6)(x + 1)”. Prepared by T CHAITANYA

  28. FACTORISATION  Example2: Q.2: Factorise: x2 – 64. Solution: Given, x2 – 64 We can also write the given expression as: ⇒ x2 – 82 [Since 8 x 8 = 64] Now by using the formula, ⇒ a2 – b2 = (a + b) (a – b) ⇒ x2 – 82 = (x + 8) (x – 8) Prepared by T CHAITANYA

  29. FACTORISATION  Example3: Q.3: Find the greatest common factor of 6x7 + 3x4 − 9x3. Solution: Given, 6x7 + 3x4 − 9x3 Now factoring the given expression, we get; ⇒ 3x3 ( 2x4 + x − 3 ) Therefore, the greatest common factor is 3x3. P r e p a r e d b y T C H A I T A N Y A

  30. FACTORISATION  Example4: Q.4: Factorise (7 x + 7 x3) + (x4 + x6). Solution: (7 x + 7 x3) + (x4 + x6) Open all the brackets. ⇒ 7x + 7x3 + x4 + x6 Taking the common factors. ⇒ 7x (1 + x2) + x4 (1 + x2) ⇒ (7x + x4) (1 + x2) ⇒ x (7 + x3 ) (1 + x2) Hence, the required factors. P r e p a r e d b y T C H A I T A N Y A

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