Application of Division Algorithm for Polynomials

Video Explanation

Question

Apply the division algorithm to find the quotient and remainder when

\[ f(x) = 10x^4 + 17x^3 – 62x^2 + 30x – 3 \]

is divided by

\[ g(x) = 2x^2 + 7x + 1. \]

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

\[ \frac{10x^4}{2x^2} = 5x^2 \]

Multiply and subtract:

\[ (10x^4 + 17x^3 – 62x^2) – (10x^4 + 35x^3 + 5x^2) = -18x^3 – 67x^2 \]

Bring down \(+30x\).

Next term

\[ \frac{-18x^3}{2x^2} = -9x \]

Multiply and subtract:

\[ (-18x^3 – 67x^2 + 30x) – (-18x^3 – 63x^2 – 9x) = -4x^2 + 39x \]

Bring down \(-3\).

Next term

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

Multiply and subtract:

\[ (-4x^2 + 39x – 3) – (-4x^2 – 14x – 2) = 53x – 1 \]

Step 3: Quotient and Remainder

Since the degree of \(53x – 1\) is less than the degree of \(2x^2 + 7x + 1\), it is the remainder.

\[ q(x) = 5x^2 – 9x – 2 \]

\[ r(x) = 53x – 1 \]

Verification

\[ (2x^2 + 7x + 1)(5x^2 – 9x – 2) + (53x – 1) = 10x^4 + 17x^3 – 62x^2 + 30x – 3 \]

Conclusion

The quotient and remainder obtained are:

\[ \boxed{q(x) = 5x^2 – 9x – 2,\quad r(x) = 53x – 1} \]