Question

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?

Answer 1

Given: f(x) = x⁴ + 2x³ – 2x² + x – 1 and divisor g(x) = x² + 2x – 3.

Step-wise calculation:

1. Let the remainder when f(x) is divided by g(x) be r(x) = ax + b.

2. Evaluate f at the roots of g:

f(1) = 1⁴ + 2 × 1³ – 2 × 1² + 1 – 1

= 1

So a + b = 1.

f(–3) = (–3)⁴ + 2(–3)³ – 2(–3)² + (–3) – 1

= 5

So –3a + b = 5.

3. Solve the system:

(–3a + b) – (a + b) = 5 – 1

⇒ –4a = 4

⇒ a = –1

Then b = 1 – a

= 1 – (–1)

= 2

4. Thus r(x) = –x + 2. To make f(x) exactly divisible by g(x), add –r(x) = x – 2.

Add x – 2 to f(x). The resulting polynomial f(x) + (x – 2) is exactly divisible by x² + 2x – 3.