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Higher Order Thinking Skills (HOTS) - Polynomials Class 9 Notes | EduRev

Created by: Indu Gupta

Class 9 : Higher Order Thinking Skills (HOTS) - Polynomials Class 9 Notes | EduRev

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Question 1. If  p(x) = x
2
â€“ 2 2 x + 1, then find  p
()
22
.
Solution: Since, p(x) = x
2
â€“ 2 2 x + 1
Then p
()
22
=
()
2
22
??
??
â€“
()
22 22 1
??
+
??
= 4 (2) â€“ 4 (2) + 1 = 8 â€“ 8 + 1 = 1
Question 2. If  a + b + c = 9, and ab + bc + ca = 26, find a
2
+ b
2
+ c
2
Solution: (a + b + c)
2
=(a
2
+ b
2
+ c
2
) + 2 (ab + bc + ca)
? [9]
2
=(a
2
+ b
2
+ c
2
) + 2 (26)  = (a
2
+ b
2
+ c
2
) + 52
? a
2
+ b
2
+ c
2
=9
2
â€“ 52 = 81â€“ 52 = 29
Question 3. Factorise : 8p
3
+
12
5
p
2
+
6
25
p

+
1
125
Solution: ? 8p
3
= (2p)
3
, and
1
125
=
3
1
5
??
??
??
? 8p
3
+
12
5
p
2
+
6
25
p

+
1
125
=  (2p)
3
+ 3 (2p)
2

1
5
??
??
??
+ 3 (2p)

23
11
55
?? ??
+
?? ??
?? ??
=
3
1
2p +
5
??
??
??
[? a
3
+ 3a
2
b + 3ab
2
+

b
3
= (a + b)
3
]
=
1
2p +
5
??
??
??

1
2p +
5
??
??
??

1
2p +
5
??
??
??
Question 4. If a, b, c are all non-zero and a + b + c = 0, prove that
22 2
ab c
3
bc ca ab
++ =
Solution: ? a + b + c = 0  ?  a
3
+ b
3
+ c
3
â€“ 3abc = 0
or a
3
+ b
3
+ c
3
= 3abc

?
33 3
abc
ab c 3
aa a a bc bc bc bc
++ = ?
22 2
ab c
3
bc ca ab
++ =
Question 5. Prove that (a + b + c)
3
â€“ a
3
â€“ b
3
â€“ c
3
=  3(a + b) (b + c) (c + a)
Solution: L.H.S. = (a + b + c)
3
â€“ a
3
â€“ b
3
â€“ c
3
=  [(a + b + c)
3
â€“ a
3
] â€“ [b
3
+ c
3
] ...(1)
(a + b + c)
3
â€“ a
3
= (b + c) [3a
2
+ b
2
+ c
2
+ 3ab + 2bc + 3ca] ...(2)
[using x
3
â€“ y
3
= (x â€“ y) (x
2
+ y
2
+ xy)]
and b
3
+ c
3
= (b + c) [b
2
+ c
2
â€“ bc]            [using x
3
+ y
3
= (x + y) (x
2
+ y
2
â€“ xy)]...(3)
From (1), (2) and (3), we get
L.H.S. = (b + c) (3a
2
+ b
2
+ c
2
+ 3ab + 2bc + 3ca) â€“ (b + c) (b
2
+ c
2
â€“ bc)
= (b + c) [3a
2
+ b
2
+ c
2
+ 3ab + 2bc + 3ca â€“ b
2
â€“ c
2
+ bc]
= (b + c) [3a
2
+ (b
2
â€“ b
2
) + (c
2
â€“ c
2
) + 3ab + (2bc + bc) + 3ca]
= (b + c) [3a
2
+ 0

+ 0 + 3ab + 3bc + 3ca]
= (b + c) [3

(a
2
+ ab

+ bc + ca)]
= 3 (b + c) [(a + b) (c + a)]
= 3 (a + b) (b + c) (c + a) = RHS
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