Page 132 - Start Up Mathematics_8 (Non CCE)

2
Coefficient of z = 1, coefficient of z = –6, constant term = –72
Split the middle term, i.e., –6 in such a way that the sum of the two parts = –6 and their
product = –72

–72 can be written as
(–6) × 12, 6 × (–12), 9 × (–8), (–9) × 8, 18 × (–4), (–18) × 4, 24 × (–3), (–24) × 3,
72 × (–1), (–72) × 1, 36 × (–2), –36 × 2
Out of these factors, only –12 + 6 = –6
2
2
\ z – 6z – 72 = z + (–12 + 6)z – 72
2
= z – 12z + 6z – 72
= z(z – 12) + 6(z – 12)

= (z + 6)(z – 12) {Taking (z – 12) common}
= (3x + 2y + 6)(3x + 2y – 12) (Putting the value of z)

EXERCISE 7.5
Factorize the following algebraic expressions by splitting the middle term:
2
2
2
(a) x – 21x + 108 (b) 40 + 3p – p (c) 9x – x – 20 (d) (x + 7)(x – 10) + 16
2
2
2
2
2
(e) (x – 5x) – 36 (f) x – x – 56 (g) x – 11x + 24 (h) 9a – 6a + 1
2
2
2
2
2
(i) 8x – 22xy + 15y 2 (j) 3a + 11ab + 6b (k) 2a – 17a – 30 (l) 2(3x – 4y) – 3(3x – 4y) – 2
2
2
2
(m) 2x + 7x – 4 (n) 5a + 13a + 6 (o) 3(2x – y) + 14(2x – y) + 8
Factorization of Quadratic Polynomials Using the Method of Completion of Perfect Squares
2
2
Step 1: In the quadratic polynomial ax + bx + c, (a π 0) make the coefficient of x as 1 by dividing/
multiplying the entire expression by it.
Ê coefficientof xˆ 2
Step 2: Add and subtract Á Ë 2 ˜ .
¯
Step 3: Write the first three terms as the square of a binomial and simplify the last two terms.
2
2
Step 4: Factorize using p – q = (p + q)(p – q).
2
Example 14: Factorize by completing the squares: (a) 2x + 6x + 1 (b) 12 – 4x – x 2
2
Solution: (a) 2x + 6x + 1
Taking 2 common from all the terms, we have
¸
Ô
Ê 1ˆ Ï Ê 3ˆ Ê ˆ 2 Ê 3ˆ 2 1Ô
3
2
2
2 x + 3x + 2¯ = 2 ()x + 2()x Á ˜ Á ˜ - Á ˜ + ˝
+
Ì
Á
˜
Ë
2¯
Ë
Ë
2¯ Ë ¯
2
Ó Ô
2Ô
˛
¸ ¸
Ï Ê Ô 3ˆ 2 9 1Ô Ï Ê Ô 3ˆ 2 Ê 9 1ˆ Ô
¸
= 2Ì Á x + ˜ - + ˝ = 2Ì Á x + ˜ - Á - ˜ ˝
˛
˛
Ó Ë Ô 2¯ 4 2Ô Ó Ë Ô 2¯ Ë 4 2¯ Ô
¸
2
¸
Ô
Ï Ê Ô 3ˆ 2 Ê 7ˆ Ô Ï Ê 3ˆ 2 Ê 7 ˆ Ô
= 2Ì Á x + ˜ - Á ˜ ˝ = Ì x + ˜ - Á ˜ ˝
2 Á
˛
˛
Ó Ë Ô 2¯ Ë 4¯ Ô Ó Ë Ô Ô 2 ¯ Ë 2 ¯ Ô
Ê Ï 3 7 ˆ Ê 3 7 ˆ ¸
= 2 Á Ì x + + ˜ Á x + - ˜ ˝
¯
Ó Ë 2 2 ¯ Ë 2 2 ˛
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