Application of Division Algorithm for Polynomials

Video Explanation

Question

Apply the division algorithm to find the quotient and remainder when

\[ f(x) = 15x^3 – 20x^2 + 13x – 12 \]

is divided by

\[ g(x) = 2 – 2x + x^2. \]

Solution

Step 1: Arrange the Divisor in Descending Powers

\[ g(x) = x^2 – 2x + 2 \]

Step 2: Apply 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 3: Polynomial Division

First term

\[ \frac{15x^3}{x^2} = 15x \]

Multiply and subtract:

\[ (15x^3 – 20x^2 + 13x) – (15x^3 – 30x^2 + 30x) = 10x^2 – 17x \]

Bring down \(-12\).

Next term

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

Multiply and subtract:

\[ (10x^2 – 17x – 12) – (10x^2 – 20x + 20) = 33x – 32 \]

Step 4: Quotient and Remainder

Since the degree of \(33x – 32\) is less than the degree of \(x^2 – 2x + 2\), it is the remainder.

\[ q(x) = 15x + 10 \]

\[ r(x) = 33x – 32 \]

Verification

\[ (x^2 – 2x + 2)(15x + 10) + (33x – 32) = 15x^3 – 20x^2 + 13x – 12 \]

Conclusion

The quotient and remainder obtained on dividing

\[ 15x^3 – 20x^2 + 13x – 12 \]

by

\[ 2 – 2x + x^2 \]

are:

\[ \boxed{q(x) = 15x + 10,\quad r(x) = 33x – 32} \]