Find remainder and quotient of the polynomial x⁴ -  9x² + 9 when divided by x² - 3x.

A polynomial is an algebraic expression with coefficients, variable and constants.

Answer: The remainder and quotient of the polynomial x⁴ -  9x² + 9 when divided by x² - 3x are 9 and x² + 3x respectively.

Explanation:

Given: Dividend = x⁴ -  9x² + 9, Divisor = x² - 3x

Let the quotient be q( x ) and remainder be r( x ).

⇒ (x⁴ -  9x² + 9)  ÷  (x² - 3x)

⇒ q ( x ) = x² + 3x and r( x ) = 9

Checking the answer by using Dividend = Divisor × Quotient + Remainder

⇒ x⁴ -  9x² + 9 = ( x² - 3x ) × ( x² + 3x ) + 9

Using the identity a² - b² = ( a + b) ( a - b )

⇒ x⁴ -  9x² + 9 = ( x² ) ²  - ( 3x )² + 9

LHS = RHS

Thus, the remainder is 9 and the quotient is x² + 3x.