Question:
Find numbers a and k so that x-2 is a factor of
[tex]f(x)=x^4-2ax^3+ax^2- x+k[/tex]
and
[tex]f(-1)=3[/tex]
Answer:
[tex]k = -2[/tex] and [tex]a=1[/tex]
Step-by-step explanation:
Given
[tex]f(x)=x^4-2ax^3+ax^2- x+k[/tex]
[tex]Factor:\ x - 4[/tex]
[tex]f(-1)=3[/tex]
Required
Find a and k
For [tex]f(-1)=3[/tex]
Substitute -1 for x
[tex]f(x)=x^4-2ax^3+ax^2- x+k[/tex]
[tex]f(-1) = (-1)^4 - 2a *(-1)^3 + a*(-1)^2 - (-1) + k[/tex]
[tex]f(-1) = 1 - 2a *-1 + a*1 +1 + k[/tex]
[tex]f(-1) = 1 +2a + a +1 + k[/tex]
[tex]f(-1) = 2 +3a + k[/tex]
Substitute 3 for f(-1)
[tex]3 = 2 +3a + k[/tex]
Collect Like Terms
[tex]3 - 2 = 3a + k[/tex]
[tex]1 = 3a + k[/tex]
Also:
If [tex]x - 2[/tex] is a factor, then
[tex]f(2) = 0[/tex]
Substitute 2 for x and 0 for f(x)
[tex]f(x)=x^4-2ax^3+ax^2- x+k[/tex]
[tex]0 = 2^4 - 2a * 2^3 + a * 2^2 - 2 + k[/tex]
[tex]0 = 16 - 2a * 8 + a * 4 - 2 + k[/tex]
[tex]0 = 16 - 16a + 4a - 2 + k[/tex]
[tex]0 = 16 - 12a - 2 + k[/tex]
Collect Like Terms
[tex]2 - 16 = - 12a + k[/tex]
[tex]-14 = k - 12a[/tex]
[tex]k = 12a - 14[/tex]
Substitute 12a - 14 for k in [tex]1 = 3a + k[/tex]
[tex]1 = 3a + 12a - 14[/tex]
[tex]1 = 15a - 14[/tex]
Collect Like Terms
[tex]15a = 1 + 14[/tex]
[tex]15a = 15[/tex]
Solve for a
[tex]a = \frac{15}{15}[/tex]
[tex]a=1[/tex]
Substitute 1 for a in [tex]k = 12a - 14[/tex]
[tex]k = 12 * 1 - 14[/tex]
[tex]k = 12 - 14[/tex]
[tex]k = -2[/tex]
