Multiply the given polynomials.

i. 2x; x² – 2x

ii. x⁵ – 1; x³ + 2x³ + 2

iii. 2y + 1; y² – 2y³ + 3y

The multiplication is as follows:

⇒ 2x × (x² – 2x – 1)

⇒ 2x.x² – 2x.2x – 2x.1

⇒ 2x²⁺¹ – 4x¹⁺¹ – 2x

⇒ 2x³ – 4x² – 2x

Therefore, the product = 2x³ – 4x² – 2x

“.” represents multiplication

ii. The multiplication is as follows:

⇒ (x⁵ – 1) × (x³ + 2x³ + 2)

⇒ x⁵.x³ + 2x³.x⁵ + 2.x⁵ – 1.x³ – 1.2x³ – 1.2

⇒ x⁵⁺³ + 2x^{3 + 5} +2x⁵ – x³ – 2x³ – 2

⇒ x⁸ + 2x⁸ + 5 + 2x⁵ – x³ – 2x³ – 2

⇒ 3x⁸ + 2x⁵ + 3x³ + 3

The product is = 3x⁸ + 2x⁵ + 3x³ + 3

“.” represents multiplication

iii. The multiplication is as follows:

⇒ (2y + 1) × (y² – 2y³ + 3y)

⇒ 2y.y² – 2y³.2y + 3y.2y + 1.y² – 1.2y³ + 1.3y

⇒ 2y²⁺¹ – 4y³⁺¹ + 6y¹⁺¹ + y² – 2y³ + 3y

⇒ 2y³ – 4y⁴ + 6y² + y² – 2y³ + 3y

⇒ -4y⁴ + 7y² + 3y

The product is = -4y⁴ + 7y² + 3y

“.” represents multiplication