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The polynomial 2x^3–ax^2+5bx+4b has a factor x–2 and, when divided by x+1, a remainder of –15 is obtained. Find the values of a and b. With these values of a and b, factorize the polynomial completely.

 Dec 2, 2017
 #1
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If x - 2 is a factor....then  2  is a zero

 

Thus  

 

2(2)^3  - a(2)^2  + 5b(2) + 4b  = 0

16 - 4a + 10b  + 4b  = 0

-4a + 14b  = -16

2a - 7b  = 8                   (1) 

 

And

2(-1)^3  - a(-1)^2  + 5b(-1) + 4b  = -15

-2   - a   - 5b + 4b   = -15

-a - b  =   -13   ⇒   -2a - 2b  =  -26      (2)

 

Add (1)  and (2)    and we have

 

-9b  =  -18    ⇒    b  = 2

 

So

 

-a - 2  = -13

-a  =  -11

a  = 11

 

So....the polynomial is

 

2x^3 - 11x^2 + 5(2)x + 4(2)

2x^3 - 11x^2 + 10x + 8

 

And we can write

 

2x^3  [ - 4x^2  -  7x^2 ] + 10x + 8

[2x^3 - 4x^2]  - [7x^2 - 10x - 8 ]

2x^2 (x  - 2)  -  [ 7x^2 - 10x - 8 ]

2x^2 (x - 2) - [ ( 7x  + 4) (x - 2) ]

(x - 2) (2x^2 - 7x - 4)

(x - 2) (2x + 1) ( x - 4)

 

 

cool cool cool

 Dec 2, 2017

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