Question

Please tell me answer fast I have my exam today 11:30am please help me​

Answer 1

Answer:

(2x + 7)(x - 2)(x + 1) are the factors.

Step-by-step explanation:

Given :

f(x) = 2x³ + 5x² - 11x - 14.  

2x + 7 is a factor of f(x).  

To find :

Factors of f(x) by factor theorem.  

Solution :

Let f(x) be 2x³ + 5x² - 11x - 14.  

Now, (2x + 7) = 0, x = -7/2  

[∵ Factor theorem]

 

∴ f(-7/2) will be:

⇒ 2(-7/2)³ + 5(-7/2)² -11(-7/2) - 14  

⇒ 2(-343/8) + 5(49/4) + 77/2 - 14  

⇒ -343 + 245 + 154 - 56/4  

⇒ 4/4  

⇒ 0

∴ (2x + 7) is a factor of f(x).

2x + 7(2x³ + 5x² - 11x - 14(x² - x - 2  

         2x³ + 7x²  

          _______________  

         -2x² - 11x - 14  

         -2x² - 7x  

         _______________  

                 -4x - 14  

                 -4x - 14  

         _______________  

                         0

∴ On dividing f(x) by (2x + 1), we get x² - x - 2 as quotient.

 

∴ f(x) = 2x³ + 5x² - 11x - 14

=> Dividend = Divisor × Quotient + Remainder  

⇒ (2x + 7)(x² - x - 2) + 0  

⇒ (2x + 7)(x² - 2x + x - 2)  

⇒ (2x + 7)[x(x - 2) + 1(x - 2)]  

⇒ (2x + 7)(x - 2)(x + 1)

∴ Factors of f(x) are (2x + 7), (x - 2), (x + 1).

Learn more :

Factor theorem: If f(x) is the polynomial and a is a real number. Then (x - a) is factor of f(x), f(a) = 0.  

Remainder theorem: If f(x) is the polynomial and is divided by (x - a), then the remainder is f(a).