Making a Polynomial Exactly Divisible

Video Explanation

Question

What must be added to the polynomial

\[ f(x) = x^4 + 2x^3 – 2x^2 + x – 1 \]

so that the resulting polynomial is exactly divisible by

\[ g(x) = x^2 + 2x – 3 \; ? \]

Solution

Step 1: Apply the 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: Divide \(f(x)\) by \(g(x)\)

First term

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

Subtract:

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

Bring down \(+x\).

Next term

\[ \frac{x^2}{x^2} = 1 \]

Subtract:

\[ (x^2 + x – 1) – (x^2 + 2x – 3) = -x + 2 \]

Step 3: Identify the Remainder

Since the degree of \(-x + 2\) is less than the degree of \(x^2 + 2x – 3\), it is the remainder.

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

Step 4: Find What Must Be Added

To make the polynomial exactly divisible, the remainder must be zero.

Therefore, the polynomial to be added is the negative of the remainder:

\[ x – 2 \]

Conclusion

The polynomial that must be added to

\[ x^4 + 2x^3 – 2x^2 + x – 1 \]

to make it exactly divisible by

\[ x^2 + 2x – 3 \]

is

\[ \boxed{x – 2} \]