Divide the polynomial p(x) = x³ - 3x² + 5x - 3 by the polynomial g(x) = x² - 2. Find the quotient and remainder.

Solution:

Let p(x) and g(x) be any polynomial of degree greater than or equal to one.

Let p(x) = x³ - 3x² + 5x - 3; g(x) = x² - 2.

We will use the long division method to find the quotient and remainder of the polynomial p(x) when divided by g(x).

Divisibility Check:

Dividend = Divisor × quotient + remainder

x³ - 3x² + 5x - 3 = (x² - 2) × (x - 3) + (7x - 9)

x³ - 3x² + 5x - 3 = (x³ - 3x² - 2x + 6) + (7x - 9)

x³ - 3x² + 5x - 3 = x³ - 3x² + 5x - 3

LHS = RHS

Hence by using the long division method, the quotient and remainder of the polynomial p(x) = x³ - 3x² + 5x - 3 when divided by the polynomial g(x) = x² - 2 are x - 3 and 7x - 9 respectively.

Divide the polynomial p(x) = x³ - 3x² + 5x - 3 by the polynomial g(x) = x² - 2. Find the quotient and remainder.

Summary:

The quotient and remainder when x³ - 3x² + 5x - 3 is divided by the polynomial g(x) = x² - 2 are x - 3 and 7x - 9 respectively.