Page 60 - SM inner class 8.cdr

Step 4. Factors of 18 are: 1 18´ , 2 ´ 9 and Required factors of 30 are (-6 ) and (-5 ).
3 ´ 6 one pair from the above \ x - 11 x + 30 = x 2 + {(- 6) + (- 5)} + 30
2
factors is taken whose sum is a.
5
= x 2 - x - x + 30
6
Step 5. The required numbers are 3 and 6.
6
(
= x x - ) - ( x - )
5
6
2
(
\ x + 9 x + 18 = x 2 + 6 + 3) x + 18
= (x - ) (x - )
5
6
3
6
= x 2 + x + x + 18 Example 16. Factorsise x + x - 12
2
= x x + ) + 3 x ( + ) Solution: Find two numbers whose product is
6
6
(
Taking out x common from first two terms and (-12 ) and sum is 1.
3 common from last two terms) Since the product is negative, one number
= (x + ) (x + ) will be positive and the other number will be
3
6
negative. Since the sum is positive, the
2
Example 15. Factorsise x - 11 x + 30
numerically greater of two numbers will be
Solution: Let us first find two numbers whose positive.
product is 30 and sum is (-11 ). So, the required factors are 4 and (-3 )
Since the product is positive therefore, either x + x - 12 = x 2 + 4 + -3)] x -12
2
(
[
both the numbers will be positive or negative. = x 2 + x - x -12
3
4
But the sum is negative, hence, both the
(
4
4
3
numbers will also be negative. = x x + ) - ( x + )
= (x + ) (x - )
4
3
E xercise 7.3
1. Factor ize the quadratic trinomials:
2
2
2
2
(i) x + x 5 + 6 (ii) x - 23 x + 42 (iii) x + 12 x + 27 (iv) x + 15 x + 56
2
2
2
2
(v) x + 19 x + 60 (vi) x + 13 x + 40 (vii) x - 10 x + 24 (viii) x + 23 x + 42
2
2
2
2
(ix) x - 17 x 16 (x) x - 21 x + 90 (xi) x - 22 x 117 (xii) x - x 9 + 20
+
+
2
2
2
(xiii) x + x 132 (xiv) x - x 5 - 24 (xv) 3x + 10x + 8
-
Summary
m Factorisation is a process of writing the given expression as a product of its factors.
m A factor which cannot be further expressed as product of factors is an irreducible factor.
m Expressions which can be transformed into the form:
2
2
2
2
2
2
2
+
;
+
;
a + 2 ab b a - 2 ab b a - b and x + ( a b x + ab can be factorised by using identities.
)
+
2
)
+
m If the given expression is of the form x + ( a b x + ab, then its factorisation is (x + a )(x + ) b
Review Exercise
1. Factor ize each of the follow ing expres sion.
2
2
(i) -5x 2 + 5xy -5x (ii) p qr + pq r + pqr 2
3
3
3
2
(iii) -4p 5 -16p q -20p q 2 (iv) p qr + 4 pq + 19 p 3
2. Factor ize using the formula of differ ence of two squares:
3
2
2
4
2
(i) 25x - 36x y 2 (ii) 49m - n 4 (iii) 81a - b 2 (iv) x - 25 x
3. Find the values of:
2
2
2
8
6
3
3
2
3
(i) ( )64 - ( ) 2 (ii) (8 2 / ) - (3 1 / ) 2 (iii) ( )42 - ( ) 2
4. Factorization by split ting the middle term:
2
2
2
2
(i) a + 10 a 24 (ii) x + x 9 + 18 (iii) p - 10 q 21 (iv) x - x 4 - 32
+
+
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Mathematics In Focus - 8