Question

If 4 is a zero of the cubic polynomial x³ – 3x² – 10x + 24, find its other two zeros.

Answer 1

Given: Polynomial f(x) = x³ – 3x² – 10x + 24 and 4 is a zero.

Step-wise calculation:

1. Verify and divide by (x – 4) using synthetic division:

Coefficients: 1 –3 –10 24 Bring down 1.

Multiply 4 × 1 = 4; add to –3 → 1. 

Multiply 4 × 1 = 4; add to –10 → −6. 

Multiply 4 × (–6) = –24; add to 24 → 0. 

So quotient is x² + x – 6 and remainder 0 (confirms 4 is a root).

2. Solve the quadratic x² + x – 6 = 0:

Discriminant = 1² – 4(1)(–6) 

= 1 + 24

= 25

Roots = `(-1 ± sqrt(25))/2` 

= `(-1 ± 5)/2` → 2 and –3

The other two zeros are 2 and –3.