An important question from factorisation chapter in which we are to Show that 2x+7 is a factor of 2x³+5x²-11x-14 and eventually we need to factorize completely.
ML Aggarwal, Avichal Publication, chapter 6, Factorisation, Question no 14
Rajan@2021Guru
Show that 2x+7 is a factor of 2x³+5x²-11x-14, hence factorise completely using factor theorem
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If 2x + 7 in factor of 2x3 + 5x2 – 11x – 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 + 77/2 – 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)