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arsen [322]
3 years ago
11

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

Mathematics
1 answer:
Slav-nsk [51]3 years ago
3 0

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

Step-by-step 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 = \frac{15}{15}

a=1

Substitute 1 for a in k = 12a - 14

k = 12 * 1 - 14

k = 12 - 14

k = -2

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