Question

Use factor theorem to factorise the following polynomials completely: x³ – 19x – 30

Answer 1

f(x) = x3 – 19x – 30
Let x = –2, then
f(–2) = (–2)³ – 19(–2) – 30
= –8 + 38 – 30
= 38 – 38
= 0
∴ (x + 2) is a factor of f(x)
Now dividing f(x) by (x + 2), we get
f(x) = x³ – 19x – 30
= (x + 2)(x² – 2x – 15)
= (x + 2){(x² –5x + 3x – 15}
= (x + 2){x(x – 5) + 3(x – 5)}
= (x + 2)(x – 5)(x + 3)

`x + 2")"overline(x^3 - 19x - 30)("x^2 - 2x - 15`
           x³ + 2x²
        –      –                   
          –2x² – 19x
          –2x²  – 4x
          +       +              
                  –15x – 30
                  –15x – 30
                 +       +      
                       x