Page 53 - SM inner class 8.cdr

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Example 23. Divide 6x - 31x + 47 by 2x - 5 \ Quotient = 3x - 8 and remainder = 7.
and verify that Dividend = Divisor ´ Quotient
+ Remainder Verification

Solution: Arranging the dividend and the Divisor ´ Quotient + Remainder
5
´
divisor in descending order of degree we get = (2x - ) (3x - ) + 7
8
2
¸
(6x - 31x + 47 ) (2x - ) 5 = 2x ( 3x - 8) - 5 3x - 8) + 7
(
3x - 8 = 6x 2 -16x -15x + 40 + 7
2
2x - 5 6x - 31x + 47 = 6x 2 - 31x + 47 = Dividend
2
_6x m 15x Hence,
_ 16x + 47 Divisor ´ Quotient +Remainder
_16x + 40 Hence verified.
-
7
E xercise 6.3

Divide the following:
3
1. 6x by 3x 2
4
3
2. 6x - 24x - 15x 2 + 3 by (3x 2 )
3. -12x + 22x 2 -16x 3 + 4 by 2x
3
4. (x - by (x +1 )
) 1
4
2
2
3
+
5. (a + 12 a 35 ) by (a + 7 ) 6. (2x + x - 27x - 36x + 155 ) by (2x - ) 5
2
2
3
7. (2m - 5m + 8m - ) 5 by (2m - ) 3 8. (x - 7 xy 18 y 2 ) by (x -9 ) y
-
3
9. (9x + 6x + ) 1 by (3x + ) 1
10. Divide and in each case verify the rela tion ®
Divi dend = Divi sor ´ Quotient ´ Remain der
3
2
3
2
(i) (6x + 5x + ) 4 by (2x + ) 1 (ii) (9a + 3a - 5a + ) 7 by (3a - ) 1
Standard Identities. Proof: (a + b ) (a - ) b
(
(
There are some important Identities that must = a a - b) + b a - b)
be remembered while working with algebraic = a 2 - ab + ab - b 2
expressions. These Identities makes our = a 2 -b 2
calculation much easier and faster. Identity 4. (x + a ) (x + ) b = x + (a + ) b x + ab
2
2
Identity 1. (a + ) b 2 = a + 2 ab + b 2 Proof: (x + a ) (x + ) b
Proof: (a + ) b 2 = (a + b ) (a + ) b = (x + )x + (x + )b
a
a
= a a + b) + b a + b) = x 2 + ax + bx + ab
(
(
= a 2 + ab + ab + b 2 = x 2 + a + b x + ab
)
(
2
= a 2 + ab + b 2 = x 2 + (algebraic sum
2
Identity 2. (a - ) b 2 = a - 2 ab + b 2 of a and b) ´ + (the product of a and b).
Proof: (a - ) b 2 = (a - b ) (a - ) b Uses of Identities
(
= a a - b) - b a - b)
(
Identities are used in solving problems
= a 2 - ab - ab + b 2 involving binomial expressions as they give us
= a 2 - ab + b 2 convenient way. Observe the use of identities
2
2
Identity 3. (a + b ) (a - ) b = a - b 2 in these examples.
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Mathematics In Focus - 8