Application of Division Algorithm for Polynomials

Video Explanation

Question

Apply the division algorithm to find the quotient and remainder when

\[ f(x) = x^3 – 6x^2 + 11x – 6 \]

is divided by

\[ g(x) = x^2 + x + 1. \]

Solution

Step 1: Write the Division Algorithm

By the division algorithm for polynomials,

\[ f(x) = g(x)\,q(x) + r(x), \]

where the degree of \(r(x)\) is less than the degree of \(g(x)\).

Step 2: Divide the Polynomials

Divide \[ x^3 – 6x^2 + 11x – 6 \] by \[ x^2 + x + 1. \]

First term

\[ \frac{x^3}{x^2} = x \]

Multiply and subtract:

\[ (x^3 – 6x^2 + 11x – 6) – (x^3 + x^2 + x) = -7x^2 + 10x – 6 \]

Next term

\[ \frac{-7x^2}{x^2} = -7 \]

Multiply and subtract:

\[ (-7x^2 + 10x – 6) – (-7x^2 – 7x – 7) = 17x + 1 \]

Step 3: Identify Quotient and Remainder

Since the degree of \(17x + 1\) is less than the degree of \(x^2 + x + 1\), this is the remainder.

Thus,

\[ q(x) = x – 7 \]

\[ r(x) = 17x + 1 \]

Verification

\[ g(x)q(x) + r(x) = (x^2 + x + 1)(x – 7) + (17x + 1) \]

\[ = x^3 – 6x^2 + 11x – 6 = f(x) \]

Conclusion

The quotient and remainder obtained on dividing \[ x^3 – 6x^2 + 11x – 6 \] by \[ x^2 + x + 1 \] are:

\[ \boxed{q(x) = x – 7,\quad r(x) = 17x + 1} \]