Factorise:(i) 4x² + 9y² + 16z² + 12xy – 24yz – 16xz(ii) 2x² + y² + 8z² – 2√2xy + 4√2yz – 8xz

(i) 4x 2 + 9y 2 + 16z 2 + 12xy - 24yz - 16xz = (2x) 2 + (3y) 2 + (-4z) 2 + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x) = (2x + 3y - 4z) 2 [∵ (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2yz + 2zx] Hence, 4x 2 + 9y 2 + 16z 2 + 12xy - 24yz - 16xz = (2x + 3y - 4z)(2x + 3y - 4z) (ii) 2x 2 + y 2 + 8z 2 - xy + yz - 8xz [∵ (a + b + c) 2 = (a) 2 + (b) 2 + (c) 2 + 2ab + 2bc + 2ca] Hence, 2x 2 + y 2 + 8z 2 -xy + yz - 8zx = (-x + y + z)(-x + y + z)

Mathematics

Factorise:
(i) 4x² + 9y² + 16z² + 12xy - 24yz - 16xz
(ii) 2x² + y² + 8z² - $2\sqrt{2}$ xy + $4\sqrt{2}$ yz - 8xz

Polynomials

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Answer

(i) 4x² + 9y² + 16z² + 12xy - 24yz - 16xz

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

= (2x + 3y - 4z)² [∵ (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx]

Hence, 4x² + 9y² + 16z² + 12xy - 24yz - 16xz = (2x + 3y - 4z)(2x + 3y - 4z)

(ii) 2x² + y² + 8z² - $2\sqrt{2}$ xy + $4\sqrt{2}$ yz - 8xz

[∵ (a + b + c)² = (a)² + (b)² + (c)² + 2ab + 2bc + 2ca]

$= (-\sqrt{2}x)^2 + (y)^2 + (2\sqrt{2}z)^2 + 2(-\sqrt{2}x)(y) + 2(y)(2\sqrt{2}z) + 2(2\sqrt{2}z)(-\sqrt{2}x) \\[1em] = (-\sqrt{2}x + y + 2\sqrt{2}z)^2$

Hence, 2x² + y² + 8z² - $2\sqrt{2}$ xy + $4\sqrt{2}$ yz - 8zx = (- $\sqrt{2}$ x + y + $2\sqrt{2}$ z)(- $\sqrt{2}$ x + y + $2\sqrt{2}$ z)

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Mathematics

Factorise:
(i) 4x² + 9y² + 16z² + 12xy - 24yz - 16xz
(ii) 2x² + y² + 8z² - $2\sqrt{2}$ xy + $4\sqrt{2}$ yz - 8xz

Polynomials

Answer

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Mathematics

Factorise:(i) 4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz(ii) 2x2 + y2 + 8z2 - 222\sqrt{2}22​xy + 424\sqrt{2}42​yz - 8xz

Polynomials

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Factorise:
(i) 4x² + 9y² + 16z² + 12xy - 24yz - 16xz
(ii) 2x² + y² + 8z² - $2\sqrt{2}$ xy + $4\sqrt{2}$ yz - 8xz

Factorise the following using appropriate identities:
(i) 9x² + 6xy + y²
(ii) 4y² - 4y + 1
(iii) $x^2 - \Big(\dfrac{y^2}{100}\Big)$

Expand each of the following, using suitable identities:
(i) (x + 2y + 4z)²
(ii) (2x - y + z)²
(iii) (-2x + 3y + 2z)²
(iv) (3a - 7b - c)²
(v) (-2x + 5y - 3z)²
(vi) $\Big[\dfrac{1}{4}a - \dfrac{1}{2}b + 1\Big]^2$

Write the following cubes in expanded form:
(i) (2x + 1)³
(ii) (2a - 3b)³
(iii) $\Big[\dfrac{3x}{2} + 1\Big]^3$
(iv) $\Big[x - \dfrac{2}{3}y\Big]^3$

Evaluate the following using suitable identities:
(i) (99)³
(ii) (102)³
(iii) (998)³