Making a Polynomial Exactly Divisible

Video Explanation

Question

What must be subtracted from the polynomial

\[ f(x) = x^4 + 2x^3 – 13x^2 – 12x + 21 \]

so that the resulting polynomial is exactly divisible by

\[ g(x) = x^2 – 4x + 3 \; ? \]

Solution

Step 1: Assume the Remainder

Since the divisor is a quadratic polynomial, the remainder is of the form

\[ r(x) = ax + b \]

Step 2: Use the Zeros of the Divisor

\[ x^2 – 4x + 3 = (x – 1)(x – 3) \]

So, \(x = 1\) and \(x = 3\) are zeros of the divisor.

Step 3: Substitute \(x = 1\)

\[ f(1) = 1 + 2 – 13 – 12 + 21 = -1 \]

\[ r(1) = a + b = -1 \quad \text{(1)} \]

Step 4: Substitute \(x = 3\)

\[ f(3) = 81 + 54 – 117 – 36 + 21 = 3 \]

\[ r(3) = 3a + b = 3 \quad \text{(2)} \]

Step 5: Solve the Equations

Subtract (1) from (2):

\[ 2a = 4 \Rightarrow a = 2 \]

Substitute in (1):

\[ 2 + b = -1 \Rightarrow b = -3 \]

Step 6: Find the Required Polynomial

\[ r(x) = 2x – 3 \]

To make the polynomial exactly divisible, this remainder must be subtracted.

Conclusion

The polynomial that must be subtracted is:

\[ \boxed{2x – 3} \]