Question

Show that 2x + 7 is a factor of 2x³ + 5x² – 11x – 14. Hence factorise the given expression completely, using the factor theorem. 

Answer 1

Let 2x + 7 = 0,
then 2x = -7
x = `(-7)/(2)`
substituting the value of x in f(x),
f(x) = 2x3 + 5x2 – 11x – 14

`f(-7/2) = 2(-7/2)^3 + 5 (-7/2)^2 -11(-7/2) -14`

= `(-343)/(4) + (245)/(4) + (77)/(2) - 14`

= `(-343 + 245 + 154 - 56)/(4)`

= `(-399 + 399)/(4)`
= 0
Hence, (2x + 7) is a factor of f(x)
Proved.
Now, 2x³ + 5x² – 11x – 14
= (2x + 7)(x² – x – 2)
= (2x + 7)[x² – 2x + x – 2]
= (2x + 7)[x(x – 2) + 1(x – 2)]
= (2x + 7)(x + 1)(x – 2)

`2x + 7")"overline(2x^3 + 5x^2  – 11x  – 14)("x^2 –  x  – 2`
             2x³ +  7x²              
             –    –                       
                  – 2x²  – 11x
                  – 2x²  – 7x
                   +       +              
                            – 4x – 14
                            – 4x – 14
                            +     +       
                                x