Question

Divide polynomial \( x^4 - 2x^3 - x + 2 \) by the polynomial \( x^2 - 3x + 2 \).

Hint:

Use long division method carefully to divide polynomials step by step.

Answer 1

Perform polynomial division:
\[ \frac{x^4 - 2x^3 - x + 2}{x^2 - 3x + 2}. \] Dividing the first term:
\[ \frac{x^4}{x^2} = x^2. \] Multiply:
\[ x^2(x^2 - 3x + 2) = x^4 - 3x^3 + 2x^2. \] Subtract:
\[ (-2x^3 - x + 2) - (-3x^3 + 2x^2) = x^3 - 2x^2 - x + 2. \] Dividing the first term:
\[ \frac{x^3}{x^2} = x. \] Multiply:
\[ x(x^2 - 3x + 2) = x^3 - 3x^2 + 2x. \] Subtract:
\[ (-2x^2 - x + 2) - (-3x^2 + 2x) = x^2 - 3x + 2. \] Divide:
\[ \frac{x^2}{x^2} = 1. \] Multiply:
\[ 1(x^2 - 3x + 2) = x^2 - 3x + 2. \] Subtract:
\[ (x^2 - 3x + 2) - (x^2 - 3x + 2) = 0. \] Thus, the quotient is:
\[ x^2 + x + 1. \]