Question

x³ - 172x - 336 = 0

Explanation:

Step1: Try rational roots

By Rational Root Theorem, possible roots are factors of 336 over factors of 1, i.e., ±1, ±2, ±3, ±4, ±6, ±7, ±8, ±12, ±14, ±16, ±21, ±24, ±28, ±42, ±48, ±56, ±84, ±112, ±168, ±336.
Test \(x = 14\): \(14^3 - 172\times14 - 336 = 2744 - 2408 - 336 = 0\). So \(x - 14\) is a factor.

Step2: Perform polynomial division

Divide \(x^3 - 172x - 336\) by \(x - 14\) using long division or synthetic division.
Synthetic division with root 14:
Coefficients: 1 (x³), 0 (x²), -172 (x), -336 (constant)
Bring down 1. Multiply by 14: 14. Add to next coefficient: 0 + 14 = 14.
Multiply 14 by 14: 196. Add to -172: 196 - 172 = 24.
Multiply 24 by 14: 336. Add to -336: 336 - 336 = 0.
So quotient is \(x^2 + 14x + 24\).

Step3: Factor quadratic

Factor \(x^2 + 14x + 24\): find two numbers that multiply to 24 and add to 14. They are 12 and 2.
So \(x^2 + 14x + 24 = (x + 12)(x + 2)\).

Step4: Write all roots

The polynomial factors as \((x - 14)(x + 12)(x + 2)\). Set to zero: \(x - 14 = 0\) or \(x + 12 = 0\) or \(x + 2 = 0\). So roots are \(x = 14\), \(x = -12\), \(x = -2\).

Answer:

The roots are \(x = -12\), \(x = -2\), \(x = 14\)