Application of Division Algorithm for Polynomials

Video Explanation

Question

Apply the division algorithm to find the quotient and remainder when

\[ f(x) = 4x^3 + 8x^2 + 8x + 7 \]

is divided by

\[ g(x) = 2x^2 – x + 1. \]

Solution

Step 1: Division Algorithm

By the division algorithm,

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

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

Step 2: Polynomial Division

First term

\[ \frac{4x^3}{2x^2} = 2x \]

Multiply and subtract:

\[ (4x^3 + 8x^2 + 8x) – (4x^3 – 2x^2 + 2x) = 10x^2 + 6x \]

Bring down \(+7\).

Next term

\[ \frac{10x^2}{2x^2} = 5 \]

Multiply and subtract:

\[ (10x^2 + 6x + 7) – (10x^2 – 5x + 5) = 11x + 2 \]

Step 3: Quotient and Remainder

Since the degree of \(11x + 2\) is less than the degree of \(2x^2 – x + 1\), it is the remainder.

\[ q(x) = 2x + 5 \]

\[ r(x) = 11x + 2 \]

Verification

\[ (2x^2 – x + 1)(2x + 5) + (11x + 2) = 4x^3 + 8x^2 + 8x + 7 \]

Conclusion

The quotient and remainder obtained are:

\[ \boxed{q(x) = 2x + 5,\quad r(x) = 11x + 2} \]