What must be subtracted from the polynomial f(x) = x⁴ + 2x³ − 13x² − 12x + 21 so that the resulting polynomial is exactly divisible by x² − 4x + 3

Video Explanation

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Given

f(x) = x⁴ + 2x³ − 13x² − 12x + 21

Divisor = x² − 4x + 3

To Find

The polynomial that must be subtracted from f(x) so that it becomes exactly divisible by x² − 4x + 3.

Concept Used

According to the division algorithm,

f(x) = g(x) · q(x) + r(x)

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

Solution

Step 1: Divide f(x) by x² − 4x + 3

x⁴ + 2x³ − 13x² − 12x + 21 ÷ (x² − 4x + 3)

First term:

x⁴ ÷ x² = x²

Multiply:

x²(x² − 4x + 3) = x⁴ − 4x³ + 3x²

Subtract:

(x⁴ + 2x³ − 13x²) − (x⁴ − 4x³ + 3x²)

= 6x³ − 16x²

Bring down −12x:

6x³ − 16x² − 12x

Next term:

6x³ ÷ x² = 6x

Multiply:

6x(x² − 4x + 3) = 6x³ − 24x² + 18x

Subtract:

(6x³ − 16x² − 12x) − (6x³ − 24x² + 18x)

= 8x² − 30x

Bring down +21:

8x² − 30x + 21

Next term:

8x² ÷ x² = 8

Multiply:

8(x² − 4x + 3) = 8x² − 32x + 24

Subtract:

(8x² − 30x + 21) − (8x² − 32x + 24)

= 2x − 3

Step 2: Identify the Remainder

Remainder r(x) = 2x − 3

Step 3: Find the Required Polynomial

To make the given polynomial exactly divisible by x² − 4x + 3, we must subtract the remainder.

Required polynomial = r(x)

= 2x − 3

Final Answer

The polynomial that must be subtracted is:

2x − 3

Conclusion

Hence, if 2x − 3 is subtracted from the polynomial f(x) = x⁴ + 2x³ − 13x² − 12x + 21, the resulting polynomial becomes exactly divisible by x² − 4x + 3.