Page 52 - SM inner class 8.cdr

Verification: We know Here we see that
Dividend = Divisor ´ Quotient 3 2
3x + 7x + 4x + 5
2
2
3
) 1
=
i.e.(8x - 6x + 10x + 3 ) (4x + (2x - 2x + ) 3
2
3
4
3x - 2 9x + 15x - 2x + 7x - 8
L.H.S = 8x 3 - 6x 2 + 10x + 3
4
R.H.S. = (4x + ) (2x 2 - 2x + ) _ 9x m 6x 3
1
3
2
3
(
= 4x ( 2x 2 - 2x + 3) + 1 2x 2 - 2x + 3) 21x - 2x + 7x - 8
= 8x 3 - 8x 2 + 12x + 2x 2 - 2x + 3 _21x m 14x 2
3
= 8x 3 - 6x 2 + 10x + 3 = L.H.S 12x + 7x - 8
2
Example 20. Using division show that (3x + ) 1 _12x m 8x
2
3
2
is a factor of 12x - 2x - x + 1 by 3x + 1
_15x - 8
2
4x - 2x + 1
_15x m 10
2
3
3x + 1 12x - 2x + x + 1
2
3
_12x ± 4x 2
4
2
3
9x + 15x - 2x + 7x - 8 = ( 3x - 2)
-6x 2 + x + 1
2
3
(3x + 7x + 4x + ) 5 + 2
m 6x 2 m 2x 3 2
5
2
R.H.S. = (3x - ) (3x + 7x + 4x + ) + 2
3x + 1 3 2
= 3 3x + 7x + 4x + 5) - 2
(
x
2
3
_ 3x ± 1 (3x + 7x + 4x + 5 + ) 2
3
2
4
3
0 9x + 21x + 12x + 15x - ( 6x + 14x 2
+ 8x + 10) + 2
Remainder is zero
2
= 9x 4 + 21x 3 + 12x 2 + 15x - 6x 3 -14x
2
3
So, (3x + is a factor of (12x - 2x + x + ) 1 -8x -10 + 2
) 1
Division of a polynomial by a polynomial = 9x 4 + 15x 3 - 2x 2 + 7x - 8 = L.H.S
having non zero Remainder: In other word, Dividend = Divisor ´ Quotient
+ Remainder
In case of polynomials it is difficult to identify 3 2 4
which polynomial is greater and which Example 22. Divide (4x - 2x + x + 10x - 25)
polynomial is smaller so, we compare the by (x + 5 )
degree of the divisor and the remainder. Solution: Arranging the terms of dividend in
Same way, we continue the division of a descending order of degree we get
4
3
2
¸
polynomial by a polynomial until we get the (x + 4 x - 2 x - 10 x - 25 ) (x + ) 5
remainder with degree less than that of the x - x + 3 x - 5
2
3
divisor.
3
4
2
x + 5 x + 4 x - 2 x + 10 x - 25
3
2
Example 21. Divide: 3x ( 5x + 3x + 2) 4 3
- (2x 2 + 8 - ) by (- +2 3x ) and check your _ x ± 5 3 x 2
x
2
answer. -x - x
m x 3 m x 2
5
Solution: Since the given dividend and the
2
divisor are not in the standard form let as 3x + 10x
2
write these in the standard form. _ 3x ± 15x
x
Dividend = 3x ( 5x 2 + 3x 3 + 2) - (2x 2 + 8 - ) - 5x - 25
=15x 3 + 9x 4 + 6x - 2x 2 - 8 + x m 5x m 25
= 9x 4 + 15x 3 + 7x - 2x 2 - 8 ´
= 9x 4 + 15x 3 - 2x 2 + 7x - 8 3 2
\ Quotient = x - x + x - 5 and remainder
3
Divisor = 3x - 2 = 0.
Now let us divide:
52
Mathematics In Focus - 8