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Polynomials

Short Answer Type

211. If x + y + z = 0, show that 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:

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

25a² - 35a + 12
= 25a² - 20a - 15a + 12
= 5a(5a - 4) - 3(5a - 4)
= (5a - 4)(5a - 3)
∴ The possible expressions for the length and breadth of the rectangle are 5a - 3 and 5a - 4.

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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?

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

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218. Evaluate 105 × 106 without direct multiplying.

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219. Factorise:

49a² + 70ab + 25b²

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220. Factorise:

25 over 4 x squared minus y cubed over 9.

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