Question

Find number a and k so that x-2 is afactor
OF F(x)=x4-2ax tax-
X+k and F(-1)=3​

Answer 1

Question:

Find numbers a and k so that x-2 is a factor of

`f(x)=x^4-2ax^3+ax^2- x+k`

and

`f(-1)=3`

Answer:

`k = -2` and
`a=1`

Explanation:

Given

`f(x)=x^4-2ax^3+ax^2- x+k`

`Factor:\ x - 4`

`f(-1)=3`

Required

Find a and k

For
`f(-1)=3`

Substitute -1 for x

`f(x)=x^4-2ax^3+ax^2- x+k`

`f(-1) = (-1)^4 - 2a *(-1)^3 + a*(-1)^2 - (-1) + k`

`f(-1) = 1 - 2a *-1 + a*1 +1 + k`

`f(-1) = 1 +2a + a +1 + k`

`f(-1) = 2 +3a + k`

Substitute 3 for f(-1)

`3 = 2 +3a + k`

Collect Like Terms

`3 - 2 = 3a + k`

`1 = 3a + k`

Also:

If
`x - 2` is a factor, then

`f(2) = 0`

Substitute 2 for x and 0 for f(x)

`f(x)=x^4-2ax^3+ax^2- x+k`

`0 = 2^4 - 2a * 2^3 + a * 2^2 - 2 + k`

`0 = 16 - 2a * 8 + a * 4 - 2 + k`

`0 = 16 - 16a + 4a - 2 + k`

`0 = 16 - 12a - 2 + k`

Collect Like Terms

`2 - 16 = - 12a + k`

`-14 = k - 12a`

`k = 12a - 14`

Substitute 12a - 14 for k in
`1 = 3a + k`

`1 = 3a + 12a - 14`

`1 = 15a - 14`

Collect Like Terms

`15a = 1 + 14`

`15a = 15`

Solve for a

`a = (15)/(15)`

`a=1`

Substitute 1 for a in
`k = 12a - 14`

`k = 12 * 1 - 14`

`k = 12 - 14`

`k = -2`