Match the two columns. Column-II gives factors for expression given in Column-I.

Column 1 Column 2

$9 x^{2} + 6 x y + y^{2}$

$(2 x + 3 y - 4 z) (2 x + 3 y - 4 z)$

$4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z$

$(x + \frac{y}{10}) (x - \frac{y}{10})$

$27 x^{3} + y^{3} + z^{3} - 9 x y z$

$(3 x + y + z) (9 x^{2} + y^{2} + z^{2} - 3 x y - y z - 3 z x)$

$x^{2} - \frac{y^{2}}{100}$

$(3 x + y) (3 x + y)$

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Solution

Finding the factors of polynomials in Column-I and match with Column-II:
Explanation for (A):
Given that polynomial is $9 x^{2} + 6 x y + y^{2}$
We can write it as
$9 x^{2} + 6 x y + y^{2} = {(3 x)}^{2} + (2 \times 3 x \times y) + {(y)}^{2} = {(3 x + y)}^{2} = (3 x + y) (3 x + y)$ $[∵ {(a + b)}^{2} = a^{2} + 2 a b + b^{2}]$
Hence, the correct match for (A) is (IV).
Explanation for (B):
Given that polynomial is $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z$
We can write it as
$4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z = {(2 x)}^{2} + {(3 y)}^{2} + {(- 4 z)}^{2} + 2 (2 x \times 3 y) + 2 (3 y \times - 4 z) + 2 (- 4 z \times 2 x) = {(2 x + 3 y - 4 z)}^{2} = (2 x + 3 y - 4 z) (2 x + 3 y - 4 z)$ $[∵ {(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
Hence, the correct match for (B) is (I).
Explanation for (C):
Given that polynomial is $27 x^{3} + y^{3} + z^{3} - 9 x y z$
We can write it as
$27 x^{3} + y^{3} + z^{3} - 9 x y z = (3 x + y + z) ({(3 x)}^{2} + {(y)}^{2} + {(z)}^{2} - (3 x \times y) - (y \times z) - (3 z \times x)) = (3 x + y + z) (9 x^{2} + y^{2} + z^{2} - 3 x y - y z - 3 z x)$ $[∵ a^{3} + b^{3} + c^{3} - 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)]$
Hence, the correct match for (C) is (III)
Explanation for (D):
Given that polynomial is $x^{2} - \frac{y^{2}}{100}$
We can write it as
$x^{2} - \frac{y^{2}}{100} = {(x)}^{2} - {(\frac{y}{10})}^{2} = (x + \frac{y}{10}) (x - \frac{y}{10})$ $[∵ a^{2} - b^{2} = (a + b) (a - b)]$
Hence, the correct match for (D) is (II)
Hence, the correct matches are (A)-(IV), (B)-(I), (C)-(III), and (D)-(II).
Column-I Column-II
(A) $9 x^{2} + 6 x y + y^{2}$ (IV) $(3 x + y) (3 x + y)$
(B) $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z$ (I) $(2 x + 3 y - 4 z) (2 x + 3 y - 4 z)$
(C) $27 x^{3} + y^{3} + z^{3} - 9 x y z$ (III) $(3 x + y + z) (9 x^{2} + y^{2} + z^{2} - 3 x y - y z - 3 z x)$
(D) $x^{2} - \frac{y^{2}}{100}$ (II) $(x + \frac{y}{10}) (x - \frac{y}{10})$

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