Question

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Q) Show that 2x+7 is a factor of 2x³+5x²-11x-14. Hence, factorise the given expression completely, using FACTOR THEOREM.
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Answer 1

Answer:

Given Equations:-

• 2x³ + 5x² - 11x - 14

• 2x + 7

To Do:-

• Show that 2x+7 is a factor of 2x³+5x²-11x-14.

Theorem Used:-

• Factor Theorem

Solution:-

If 2x + 7 in factor of 2x³+5x²–11x–14

then on putting 2x + 7 = 0

x = $\dfrac{-7}{2}$

As P(x) = 0

⟾ P($\dfrac{-7}{2}$) = 0

= 2($\dfrac{-7}{2}$)³ + 5($\dfrac{-7}{2}$)² – 11($\dfrac{-7}{2}$) – 14

= $\dfrac{-343}{4}$ + $\dfrac{245}{4}$+ $\dfrac{77}{2}$ – 14

= $\dfrac{-399}{4}$ + $\dfrac{245+154}{4}$

= $\dfrac{-399}{4}$+ $\dfrac{399}{4}$ = 0

Hence, 2x + 7 is one factor.

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Now Factorizing 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)

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Answer 2

Answer

Let the function be represented by f(x).

As x+2, exactly divides f(x),

−2 is a zero of f(x), hence f(−2)=0

f(x)=2x³ +5x² −x − k

⇒k = 6