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Using the Factor Theorem, Show That: (Iv) 2x + 7 is a Factor 2x3 + 5x2 − 11x – 14. Hence, Factorise the Given Expression Completely. - ICSE Class 10 - Mathematics

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Question

Using the factor Theorem, show that:

(iv) 2x + 7 is a factor 2x3 + 5x2 − 11x – 14. Hence, factorise the given expression completely.

Solution

f(x) = 2x3 + 5x2 - 11x - 14
2x + 7 = 0  ⇒  x = `((-7)/2)`

Remainder = f  `((-7)/2)`

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

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

= `(-49)/2 + 77/2 -14`

= `28/2 -14`

= 14 - 14=0

Hence, (2x + 7) is a factor of f(x).
Now, we have:

∴ `2 x  ^ 3 +  5 x^2 - 11x -14 = (2x + 7 ) (x^2- x-2)`

                                               = `(2x +  7)(x^2 - 2x + x -2 )`

                                               = `(2x + 7 )[x(x-2)+(x-2)]`

                                               = `(x + 7)(x-2)(x+1)`   

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Solution Using the Factor Theorem, Show That: (Iv) 2x + 7 is a Factor 2x3 + 5x2 − 11x – 14. Hence, Factorise the Given Expression Completely. Concept: Factor Theorem.
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