NCERT Solutions for Class 9 Maths Chapter 2: Polynomials

Chapter 2.

Q. Find the value of (13)3 + (15)3 โ€“ (28)3.

Ans. (13)3 + (15)3 โ€“ (28)3 =(13)3 + (15)3 + (โ€“ 28)3
Let a = 13, b = 15 and c = โ€“ 28
a + b + c = 13 + 15 โ€“ 28 = 0
a3 + b3 + c3 = 3abc
โ‡’ (13)3 + (15)3 + (โ€“ 28)3 = 3(13) (15) (โ€“ 28)
โ‡’ (13)3 + (15)3 โ€“ (28)3 = โ€“ 3(13) (420)
(13)3 + (15)3 โ€“ (28)3 = โ€“ 16380

Q. Prove that : a3 + b3 + c3 โ€“ 3abc=1/22(a+b+c){(a-b)2+(b-c)2+(c-a)2}
Ans. Since we know that, a3 + b3 + c3 โ€“ 3abc
Q. Prove that : (a + b + c)3 โ€“ a3 โ€“ b3 โ€“ c3 = 3(a + b)(b + c) (c + a).
Ans. L.H.S. = (a + b + c)3 โ€“ a3 โ€“ b3 โ€“ c3
= {(a + b + c)3 โ€“ a3} โ€“ (b3 + c3)
= [(a + b + c โ€“ a) {(a + b + c)2 + a(a + b + c) + a2}] โ€“ (b3 + c3)
= (b + c) {a2 + b2 + c2 + 2ab + 2bc + 2ac + a2 + ab + ac + a2} โ€“ (b3 + c3)
= (b + c) (3a2 + b2 + c2 + 3ab + 2bc + 3ac) โ€“ (b3 + c3)
= (b + c) (3a2 + b2 + c2 + 3ab + 2bc + 3ac) โ€“ (b + c) (b2 โ€“ bc + c2)
= (b + c) (3a2 + b2 + c2 + 3ab + 2bc + 3ac โ€“ b2 + bc โ€“ c2)
= (b + c) (3a2 + 3ab + 3bc + 3ac)
= 3(b + c) (a2 + ab + bc + ac)
= 3(b + c) {a(a + b) + c (a + b)}
= 3 (b + c) (a + b) (c + a)
= 3 (a + b) (b + c) (c + a)
= R.H.S.
Q. Factorise : x3 + 13x2 + 32x + 20.

Ans. Let p (x) = x3 + 13x2 + 32x + 20
Put x = โ€“ 1
= (โ€“ 1)3 + 13(โ€“ 1)2 + 32(โ€“ 1) + 20
= โ€“ 1 + 13 โ€“ 32 + 20
33 โ€“ 33
= 0 (x + 1) is a factor of x3 + 13x2 + 32x + 20
Now, divide x3 + 13x2 + 32x + 20 by x + 1, we get

$$\space\space\space\space\space\space\space\space\space\space x^2+12x+20\\x+1)\overline{x^3+13x^2+32x+20}\\\space\space\space\space\space\space\space\space\space\space x^3+x^2\\\space\space\space\space\space\space\space\space\space\space (-)(-)\\\space\space\space\space\space\space\space\space\space\space \overline{12x^2+32x+20}\\\space\space\space\space\space\space\space\space\space\space 12x^2+12x\\\space\space\space\space\space\space\space\space\space\space (-)(-)\\\space\space\space\space\space\space\space\space\space\space \overline{20x+20}\\\space\space\space\space\space\space\space\space\space\space {20x+20}\\\space\space\space\space\space\space\space\space\space\space (-)(-)\\\space\space\space\space\space\space\space\space\space\space\space\space\overline{ร—}\\\underline{}$$

x3 + 13x2 + 32x + 20 = (x + 1) (x2 + 12x + 20)
= (x + 1) {x2 + 10x + 2x + 20}
= (x + 1) {x (x + 10) + 2 (x + 10)}
x3 + 13x2 + 32x + 20 = (x + 1) (x + 2) (x + 10).
Q. Factorise : 2x3 โ€“ 3x2 โ€“ 17x + 30.
Ans. p (x) = 2x3 โ€“ 3x2 โ€“ 17x + 30
Put x = 2 p (2) = 2(2)3 โ€“ 3(2)2 โ€“ 17(2) + 30
= 16 โ€“ 12 โ€“ 34 + 30
= 16 โ€“ 16 = 0
(x โ€“ 2) is a factor of p (x)

$$\space\space\space\space\space\space\space\space\space\space 2x^2+x-15\\x-2)\overline{2x^3-3x^2-17x+30}\\\space\space\space\space\space\space\space\space\space\space 2x^3-4x^2\\\space\space\space\space\space\space\space\space\space\space (-)(+)\\\space\space\space\space\space\space\space\space\space\space \overline{x^2-17x+30}\\\space\space\space\space\space\space\space\space\space\space x^2-2x\\\space\space\space\space\space\space\space\space\space\space (-)(+)\\\space\space\space\space\space\space\space\space\space\space \overline{-15x+20}\\\space\space\space\space\space\space\space\space\space\space {-15x+20}\\\space\space\space\space\space\space\space\space\space\space (+)(-)\\\space\space\space\space\space\space\space\space\space\space \overline{ร—}\\\space\space\space\space\space\space\space\space\space\space \underline{}$$

2x3 โ€“ 3x2 โ€“ 17x + 30
= (x โ€“ 2) (2x2 + x โ€“ 15)
= (x โ€“ 2) {2x2 + 6x โ€“ 5x โ€“ 15}
= (x โ€“ 2) {2x (x + 3) โ€“ 5 (x + 3)}
= (x โ€“ 2) (x + 3) (2x โ€“ 5) 2x3 โ€“ 3x2 โ€“ 17x + 30
= (x โ€“ 2) (2x โ€“ 5) (x + 3).

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