Question

of 2x-1 is the factor of 2x^3+ax^2-5x+2 b if ab=1 then find a-8b​

Answer 1

Answer:

Step-by-step explanation:

Let p (x) = 2x³ + ax² - 5x + 2b = 0

If (2x - 1 ) is the factor of p (x) ,then 2x- 1= 0 ⇒2x = 1 ⇒x = 1/2.

Now substituting " x = 1/2 " in p (x ),we get

p ( 1/2 ) = 2 (1/2)³ + a (1/2)² - 5 (1/2) + 2b = 0

⇒2 (1/8) + a (1/4) - 5/2 + 2b = 0

⇒ 1/4 + a / 4 - 5/2 + 2b = 0

⇒( 1 + a -10 + 8b ) / 4 = 0

⇒ a + 8b - 9 = 0

⇒ a + 8b = 9------eq ( 1 )

Given that, ab = 1 ⇒ a = 1/b

Now substituting " a = 1/b " in eq ( 1 ) , we get

⇒1/b + 8b = 9

⇒1 + 8b² / b = 9

⇒ 1 + 8b² = 9b

⇒8b² - 9b + 1 = 0

⇒8b² - 8b - b + 1 = 0

⇒ 8b ( b - 1 ) - 1 ( b - 1 ) = 0

⇒(b - 1 ) ( 8b - 1 ) = 0

⇒b - 1 = 0 (or) 8b - 1 = 0

⇒ b = 1 (or) b = 1/8

Now substituting " b = 1 or 1/8 " in eq (1),we get

a+ 8b = 9

⇒a + 8 × 1 = 9 (or ) a + 8 × 1/8 = 9

⇒a + 8 = 9 (or) a + 1 = 9

⇒a = 9 - 8 (or) a = 9 - 1

⇒a = 1 (or) a = 8

∴ a - 8b = 1 - 8 (1) 1 ' 8 = - 7 is the answer. ( If a = 1 and b = 8 )

= 8 - 8 (1/8) = 8 - 1 = 7 is the answer. (If a= 8 and b = 1/8)