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Polynomials

Short Answer Type

161.

Simplify: (x + y)³ - (x - y)³ - 6y(x²- y²)

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162. If x² + px + q = (x + a)(x + b), then factorise x² + pxy + qy².

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163. Prove that: (a + b)³ + (b + c)³ + (c + a)³ - 3(a + b)(b + c)(c + a) = 2(a³ + b³+ c³ - 3abc).

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164. Evaluate using suitable identify: (999)³.

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165. Multiply 9x² + 25y² + 15xy + 12x -20y + 16 by 3x - 5y - 4 using suitable identity.

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166. Show that if a + b is not zero, then the equation a(x - a) = 2ab - b(x - b) has a solution x = a + b.

a(x - a) = 2ab - b(x - b)
⇒ ax - a² = 2ab - bx + b²⇒ ax + bx = a² + 2ab + b²∴ x(a + b) = (a + b)² | Using Identity I
∴ x = a + b
| Cancelling (a + b) from both sides as (a + b) ≠ 0
Hence the given equation has a solution x = a + b.

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167. Show that if 2(a² + b²) = (a + b)², then a = b.

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168.

Factorise each of the following:

a⁴ - b⁴ (b)a⁴ - 16b⁴

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169.

Factorise each of the following:

^{a⁴ - 16b⁴}

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170.

Factorise each of the following:

a² - (b - c)²

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