What must be added to the polynomial f(x) = x⁴ + 2x³ − 2x² + x − 1 so that the resulting polynomial is exactly divisible by x² + 2x − 3

Video Explanation

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Given

f(x) = x⁴ + 2x³ − 2x² + x − 1

Divisor = x² + 2x − 3

To Find

The polynomial that must be added to f(x) so that it becomes exactly divisible by x² + 2x − 3.

Concept Used

If a polynomial f(x) is divided by g(x), then

f(x) = g(x) · q(x) + r(x)

If we subtract the remainder r(x) from f(x), the resulting polynomial becomes exactly divisible by g(x).

Solution

Step 1: Divide f(x) by x² + 2x − 3

x⁴ + 2x³ − 2x² + x − 1 ÷ (x² + 2x − 3)

First term:

x⁴ ÷ x² = x²

Multiply:

x²(x² + 2x − 3) = x⁴ + 2x³ − 3x²

Subtract:

(x⁴ + 2x³ − 2x²) − (x⁴ + 2x³ − 3x²)

= x²

Bring down +x:

x² + x

Next term:

x² ÷ x² = 1

Multiply:

1(x² + 2x − 3) = x² + 2x − 3

Subtract:

(x² + x − 1) − (x² + 2x − 3)

= −x + 2

Step 2: Identify the Remainder

Remainder r(x) = −x + 2

Step 3: Find the Required Polynomial

To make the given polynomial exactly divisible by x² + 2x − 3, we must subtract the remainder.

Required polynomial = −r(x)

= −(−x + 2)

= x − 2

Final Answer

The polynomial that must be added is:

x − 2

Conclusion

Hence, if x − 2 is added to the polynomial f(x) = x⁴ + 2x³ − 2x² + x − 1, the resulting polynomial becomes exactly divisible by x² + 2x − 3.