Question

Can u help me ASAP. i need to know how to do it step by step​

Answer 1

19

Explanation:

`f(x) =4x^3-8x^2+ax+b` has a factor
`2x-1` and

when divided by
`x+2`, remainder is 20.

To find:Remainder when divided by (x-1) ?

Solution:

`2x-1` is a factor

`2x - 1 = 0 \Rightarrow x = (1)/(2)` when we put this value of x to the function, it will become 0.

i.e.

`\Rightarrow f((1)/(2)) =0 =4* (\frac{1}2)^3-8(\frac{1}2)^2+(a)/(2)+b=0\\\Rightarrow (1)/(2)-2+(a)/(2)+b=0\\\Rightarrow 1-4+a+2b=0\\\Rightarrow a +2b=3 ......(1)`

Remainder is 20 when f(x) is divided by
`x+2`

i.e.

`f(-2) =20`

`\Rightarrow f(-2) =20 =4* (-2)^3-8(-2)^2-2a+b=20\\\Rightarrow -32-32-2a+b=20\\\Rightarrow -2a+b=84 ...... (2)`

Solving (1) and (2), Multiply equation (1) by 2 and adding to (2):

`5b=6+84\\\Rightarrow b = (90)/(5) = \bold{18}`

By equation (1):

`a+2(18) = 3\\\Rightarrow a = -33`

So, the equation becomes:

`f(x) =4x^3-8x^2-33x+18`

`\Rightarrow f(1) = 4(1) -8 (1) -33(1) +18 = \bold{19}`

So, when divided by (x-1), remainder will be 19.