Question

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

Answer 1

If 2x + 7 in factor of 2x³ + 5x² - 11 x - 14 then on putting 2x + 7 = 0
x = `-(7)/(2)`
`f(-7/2)` = 0
= `2(-7/2)^3 + 5(-7/2)^2 -11(7/2)-14`
= `(-343)/(4) + (245)/(4) + (77)/(4) -14`
= `(-399)/(4) + (245 + 154)/(4)`
= `(-399 + 399)/(4)` = 0
Hence 2x + 7 is one factor.
Now 2x³ + 5x² - 11x - 14
= x² (2x + 7) -x (2x + 7) -2 (2x + 7)
= (2x + 7) (x² - x - 2)
= (2x + 7) (x² + x - 2x - 2)
= (2x + 7) [x (x + 1) -2 (x + 1)]
= (2x + 7) (x - 2) (x + 1).