14 Factorisation
14.1 Introduction
14. 1.1 Factors Of Natural Numbers
14.1.2 Factors Of Algebraic Expressions
14.2 What Is Factorization
14.2.1 Method Of Common Factors
14.2.2 Factorization By Regrouping Terms
14.2.3 Factorization Using Identities
14.2 4 Factors Of The Form (X + A)( X + B)
14.3 Division Of Algebraic Expressions
14.3.1 Division Of Monomial By Another Monomial
14.3 2 Division Of A Polynomial By A Monomial
14.4 Division Of Algebraic Expressions Continued Polynomial By Polynomial
14.5 Can You Find The Error
WHAT HAVE WE DISCUSSED?
1. When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
2. An irreducible factor is a factor which cannot be expressed further as a product of factors. 3. A systematic way of factorising an expression is the common factor method. It consists of three steps: (i) Write each term of the expression as a product of irreducible factors (ii) Look for and separate the common factors and (iii) Combine the remaining factors in each term in accordance with the distributive law.
4. Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.
5. In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.
6. A number of expressions to be factorised are of the form or can be put into the form : a2 + 2 ab + b2, a2 – 2ab + b2, a2 – b2 and x2 + (a + b) + ab. These expressions can be easily factorised using Identities I, II, III and IV, given in Chapter 9,
a2 + 2 ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
a2 – b2 = (a + b) (a – b)
x2 + (a + b) x + ab = (x + a) (x + b)
7. In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab. Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.
8. We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.
9. 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. 10. 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.
11. 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
Example 1: Factorise 12a2b + 15ab2
Example 2: Factorise 10x2 – 18x3 + 14x4
Factorise: (i) 12x + 36 (ii) 22y – 33z (iii) 14pq+35pqr
Example 3: Factorise 6xy – 4y + 6 – 9x.
Example 4: Factorise x2 + 8x + 16
Example 5: Factorise 4y2 – 12y + 9
Example 6: Factorise 49p2 – 36
Example 7: Factorise a2 – 2ab + b2 – c2
Example 8: Factorise m4 – 256
Example 9: Factorise x2 + 5x + 6
Example 10: Find the factors of y2 –7y +12.
Example 11: Obtain the factors of z2 – 4z – 12.
Example 12: Find the factors of 3m2 + 9m + 6.
Example 13: Do the following divisions.
(i) –20x4 ⎟ 10x2(ii) 7x2y2z2 ⎟ 14xyz
Divide. (i) 24xy2z3 by 6yz2(ii) 63a2b4c6 by 7a2b2c3
Example 14: Divide 24(x2yz + xy2z + xyz2) by 8xyz
Example 15: Divide 44(x4 – 5x3 – 24x2) by 11x (x – 8)
Example 16: Divide z(5z2 – 80) by 5z(z + 4)
EXERCISE 14.1
1. Find the common factors of the given terms.
(i) 12x, 36
(ii) 2y, 22xy
(iii) 14 pq, 28p2q2
(iv) 2x, 3x2, 4
(v) 6 abc, 24ab2, 12 a2b
(vi) 16 x3, – 4x2, 32x
(vii) 10 pq, 20qr, 30rp
(viii) 3x2 y3, 10x3 y2,6 x2 y2z
2. Factorise the following expressions.
(i) 7x – 42
(ii) 6p – 12q
(iii) 7a2 + 14a
(iv) – 16 z + 20 z3
(v) 20 l2 m + 30 a l m
(vi) 5 x2 y – 15 xy2
(vii) 10 a2 – 15 b2 + 20 c2
(viii) – 4 a2 + 4 ab – 4 ca
(ix) x2 y z + x y2z + x y z2
(x) a x2 y + b x y2 + c x y z
3. Factorise.
(i) x2 + x y + 8x + 8y
(ii) 15 xy – 6x + 5y – 2
(iii) ax + bx – ay – by
(iv) 15 pq + 15 + 9q + 25p
(v) z – 7 + 7 x y – x y z
EXERCISE 14.2
1. Factorise the following expressions.
(i) a2 + 8a + 16
(ii) p2 – 10 p + 25
(iii) 25m2 + 30m + 9
(iv) 49y2 + 84yz + 36z2
(v) 4x2 – 8x + 4
(vi) 121b2 – 88bc + 16c2
(vii) (l + m)2 – 4lm (Hint: Expand ( l + m)2first)
(viii) a4 + 2a2b2 + b4
2. Factorise.
(i) 4p2 – 9q2
(ii) 63a2 – 112b2
(iii) 49x2 – 36
(iv) 16x5 – 144x3
(v) (l + m)2 – (l – m)2
(vi) 9x2 y2 – 16
(vii) (x2 – 2xy + y2) – z2
(viii) 25a2 – 4b2 + 28bc – 49c2
3. Factorise the expressions.
(i) ax2 + bx
(ii) 7p2 + 21q2
(iii) 2x3 + 2xy2 + 2xz2
(iv) am2 + bm2 + bn2 + an2
(v) (lm + l) + m + 1
4. Factorise.
(i) a4 – b4
(ii) p4 – 81
(iii) x4 – (y + z)4
(iv) x4 – (x – z)4
(v) a4 – 2a2b2 + b4
5. Factorise the following expressions.
(i) p2 + 6p + 8
(ii) q2 – 10q + 21
(iii) p2 + 6p – 16
(vi) y (y + z) + 9 (y + z)
(vii) 5y2 – 20y – 8z + 2yz
(viii) 10ab + 4a + 5b + 2EXERCISE 14.3
1. Carry out the following divisions.
(i) 28x4 ➗ 56x
(ii) –36y3 ➗ 9y2
(iii) 66pq2r3 ➗ 11qr2
(iv) 34x3y3z3 ➗51xy2z3
(v) 12a8b8 ➗ (– 6a6b4)
2. Divide the given polynomial by the given monomial.
(i) (5x2 – 6x) ➗3x
(ii) (3y8 – 4y6 + 5y4) ➗ y4
(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ➗ 4x2y2z2
(iv) (x3 + 2x2 +3x) ➗ 2x
(v) (p3q6 – p6q3) ➗ p3q3
3. Work out the following divisions.(i) (10x – 25) ➗ 5
(ii) (10x – 25) ➗ (2x – 5)
(iii) 10y(6y + 21) ➗ 5(2y + 7)
(iv) 9x2y2(3z – 24) ➗ 27xy(z – 8)
(v) 96abc(3a – 12) (5b – 30) ➗ 144(a – 4) (b – 6)
4. Divide as directed.
(i) 5(2x + 1) (3x + 5) ➗ (2x + 1)
(ii) 26xy(x + 5)(y – 4) ➗ 13x(y – 4)
(iii) 52pqr (p + q) (q + r) (r + p) ➗ 104pq(q + r) (r + p)
(iv) 20(y + 4) (y2 + 5y + 3) ➗ 5(y + 4)
(v) x(x + 1) (x + 2) (x + 3) ➗ x(x + 1)
5. Factorise the expressions and divide them as directed.
(i) (y2 + 7y + 10) ➗ (y + 5)
(ii) (m2 – 14m – 32) ➗ (m + 2)
(iii) (5p2 – 25p + 20) ➗ (p – 1)
(iv) 4yz(z2 + 6z – 16) ➗ 2y(z + 8)
(v) 5pq(p2 – q2) ➗ 2p(p + q)
(vi) 12xy(9x2 – 16y2) ➗ 4xy(3x + 4y)EXERCISE 14.4
Find and correct the errors in the following mathematical statements.(a) x2 + 5x + 4 gives (– 3)2 + 5 (– 3) + 4 = 9 + 2 + 4 = 15
(b) x2 – 5x + 4 gives (– 3)2 – 5 ( – 3) + 4 = 9 – 15 + 4 = – 2
(c) x2 + 5x gives (– 3)2 + 5 (–3) = – 9 – 15 = – 24
11. (y – 3)2 = y2 – 9
12. (z + 5)2 = z2 + 25
13. (2a + 3b) (a – b) = 2a2 – 3b2
14. (a + 4) (a + 2) = a2 + 8
15. (a – 4) (a – 2) = a2 – 8
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