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We know that x 3 + y 3 + z 3 - 3xyz = (x + y + z)(x 2 + y 2 + z 2 - xy - yz - zx) | Using Identity VIII = (0)(x 2 + y 2 + z 2 - xy - yz - zx) | ∵ x + y + z = 0 = 0 ⇒ x 3 + y 3 + z 3 = 3xyz.

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211. If x + y + z = 0, show that x³ + y³ + Z³ = 3xyz.

We know that
x³ + y³ + z³ - 3xyz
= (x + y + z)(x² + y² + z²- xy - yz - zx)
| Using Identity VIII
= (0)(x² + y² + z² - xy - yz - zx)
| ∵ x + y + z = 0
= 0
⇒ x³ + y³ + z³ = 3xyz.

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212. Without actually calculating the cubes, find the value of each of the following:
(- 12)³+ (7)³ + (5)³

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213. Without actually calculating the cubes, find the value of each of the following:
(28)³ + (- 15)³ + (- 13)³

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214. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:

box enclose bold space bold Area bold space bold colon bold space bold 25 bold a to the power of bold 2 bold minus bold 35 bold a bold plus bold 12 end enclose

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215. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:

box enclose bold Area bold space bold colon bold space bold 35 bold y to the power of bold 2 bold plus bold 13 bold y bold minus bold 12 end enclose

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216.

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

$box enclose bold Volume bold space bold colon bold space bold 3 bold x to the power of bold 2 bold minus bold 12 bold x end enclose$

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217.

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

$box enclose bold Volume bold space bold colon bold space bold 12 bold ky to the power of bold 2 bold plus bold 8 bold ky bold minus bold 20 bold k end enclose$