MP Board · Class 10 · Mathematics · PolynomialsDiscuss the concept of polynomial division and explain how to divide a polynomial by another polynomial. Provide a step-by-step example of dividing $x^3 - 2x^2 - 5x + 1$ by $x - 1$.

Introduction to Polynomial Division\nPolynomial division is a process used to divide one polynomial by another to find the quotient and remainder. It is similar to numerical division but involves polynomials.

Steps for Polynomial Division

1. Divide the leading term of the dividend by the leading term of the divisor: This gives the first term of the quotient.
2. Multiply the entire divisor by this term and subtract it from the dividend: This process is repeated until the degree of the remainder is less than that of the divisor.

Example: Dividing $x^3 - 2x^2 - 5x + 1$ by $x - 1$

1. Divide $x^3$ by $x$ to get $x^2$.
2. Multiply $x - 1$ by $x^2$ to get $x^3 - x^2$ and subtract this from $x^3 - 2x^2 - 5x + 1$ to get $-x^2 - 5x + 1$.
3. Divide $-x^2$ by $x$ to get $-x$.
4. Multiply $x - 1$ by $-x$ to get $-x^2 + x$ and subtract this from $-x^2 - 5x + 1$ to get $-6x + 1$.
5. Divide $-6x$ by $x$ to get $-6$.
6. Multiply $x - 1$ by $-6$ to get $-6x + 6$ and subtract this from $-6x + 1$ to get $-5$. \nThus, the quotient is $x^2 - x - 6$ and the remainder is $-5$. \nThe division can be represented as: [ x^3 - 2x^2 - 5x + 1 = (x - 1)(x^2 - x - 6) - 5 ]