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PEDAGOGY OF MATHEMATICS
1. PEDAGOGY OF
MATHEMATICS β PART II
By
Dr. I. Uma Maheswari
Principal
Peniel Rural College of Education,Vemparali, Dindigul District
iuma_maheswari@yahoo.co.in
19. Solution:
p(x) = 3x3 β 4x2 + 7x β 5
When it is divided by x +3,
p(-3) = 3(-3)3 β 4(-3)2 + 7(-3) β 5
= 3(-27) β 4(9) β 21 β 5
= -81 β 36 β 21 β 5
= -143
The remainder is -143.
20. Solution:
p(x) = x2018 + 2018
When it is divided by x β 1,
p(1) = 12018 + 2018
= 1 + 2018
= 2019
The remainder is 2019.
21. Solution:
p(x) = 2x3 β kx2 + 3x + 10
When it is exactly divided by x β 2,
P(2) = 0
2(2)3 β k(2)2 + 3(2) + 10 = 0
2(8) β k(4) + 6 + 10 = 0
16 β k(4) + 6 + 10 = 0
16 β 4k + 6 + 10 = 0
32 β 4k = 0
32 = 4k
β΄ k = 324
= 8
The value of k = 8
22. Solution:
p(x1) = 2x3 + ax2 + 4x β 12
When it is divided by x β 3,
p(3) = 2(3)3 + a(3)2 + 4(3) β 12
= 54 + 9a + 12 β 12
= 54 + 9a β¦β¦β¦.(R1)
p(x2) = x3 + x2 β 2x + a
When it is divided by x β 3,
p(3) = 33 + 32 β 2(3) + a
= 27 + 9 β 6 + a
= 30 + a β¦β¦β¦(R2)
The given remainders are same (R1 = R2)
β΄ 54 + 9a = 30 + a
9a β a = 30 β 54
8a = -24
β΄ a = -24/8
= -3
Consider R2,
Remainder = 30 β 3
= 27
23. (i) x3 + 5x2 β 10x + 4
Solution:
p(x) = x3 + 5x2 β 10x + 4
p(1) = 13 + 5(1) β 10(1) +
4
= 1 + 5 β 10 + 4
= 10 β 10
= 0
β΄ x β 1 is a factor of p(x)
(ii) x4 + 5x2 β 5x + 1
Solution:
p(1) = 14 + 5(1)2 β 5(1) + 1
= 1 + 5 β 5 + 1
= 7 β 5
= 2
= 0
β΄ x β 1 is not a factor of
p(x)
24. Solution:
p(x) = 2x3 β 5x2 β 28x + 15
x β 5 is a factor
p(5) = 2(5)3 β 5(5)2 β 28(5) + 15
= 250 β 125 β 140 + 15
= 265 β 265
= 0
β΄ x β 5 is a factor of p(x)
25. Solution:
p(x) = x3 β 3x2 β mx + 24
when x + 3 is a factor
P(-3) = 0
(-3)3 β 3(-3)2 β m(-3) + 24 =
0
-27 β 27 + 3m + 24 = 0
-54 + 24 + 3m = 0
-30 + 3m = 0
3m = 30
m = 303
= 10
The value of m = 10
26. Solution:
p(x) = ax2 + 5x + b
when (x-2) is a factor
P(2) = 0
a(2)2 + 5(2) + b = 0
4a + 10 + b = 0
4a + b = -10 β¦β¦.(1)
when (x β 12) is a factor
p(12) = 0
a(12)2 + 5(12) + b = 0
Multiply by 4
a + 10 + 4b = 0
a + 46 = -10 β¦β¦.(2)
From (1) and (2) we get
4a + b = a + 4b
4a β a = 4b β b
3a = 3b
a = b
Hence it is proved.
27. Solution:
p(x) = kx3 β 2x2 + 25x β 26
When it is divided by x β 1
P(1) = 0
k(1)3 β 2(1)2 + 25(1) β 26 = 0
k β 2 + 25 β 26 = 0
k + 25 β 28 = 0
k β 3 = 0
k = 3
The value of k = 3
28. Solution:
Let the area of a rectangle be p(x)
p(x) = x2 β 2x β 8
When x + 2 is the side of the
rectangle
p(-2) = (-2)2 β 2(-2) β 8
= 4 + 4 β 8
= 8 β 8
= 0
When x β 4 is the side of the rectangle.
P(4) = (4)2 β 2(4) β 8
= 16 β 8 β 8
= 16 β 16
= 0
(x + 2) and (x β 4) are the sides of a
rectangle