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Use the remainder theorem and synthetic division to find f(k)

K=-1;f(x)=x2-4x+5

F(k)=

Use the remainder theorem and synthetic division to find f(k) for the given value of k.

F(x) = -x3-10×2-25x-13;k=-6

F(-6)=

Use the remainder theorem and synthetic division to find f(k) for the given value of k.

F(x)=3×4-17×3-3×2+4x+4;k=-1/3

F(-1/3)=

Use synthetic division to decide whether the given number k is a zero of the polynomial function. If it is not, give the value off(k).

F(x)=x2-8x+15;k=5

Is 5 a zero of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

For the polynomial function, use the remainder theorem and synthetic division to find f(k).

F(x)=3×2+48,k=4i

F(4i) =

Express f(x) in the form f(x)=(x-k)q(x)+r for the given value of k.

F(x)=x3+5×2+7x+2,k=-2

**Expert Solution Preview**

Introduction:

In this assignment, we will be applying the concepts of the remainder theorem and synthetic division to find the value of f(k) for given polynomial functions and values of k. These techniques are essential in understanding the behavior of polynomial functions and finding zeros.

1. Question: Use the remainder theorem and synthetic division to find f(k) for the given value of k. K = -1, f(x) = x^2 – 4x + 5.

Answer: To find f(k), we substitute the given value of k into the polynomial function f(x). Therefore, for k = -1, we have:

f(-1) = (-1)^2 – 4(-1) + 5

= 1 + 4 + 5

= 10

Hence, f(-1) = 10.

2. Question: Use the remainder theorem and synthetic division to find f(k) for the given value of k. F(x) = -x^3 – 10x^2 – 25x – 13; k = -6.

Answer: To find f(k), we can use synthetic division with the given value of k = -6:

-6 | -1 -10 -25 -13

| 6 24 6

|———————–

-1 -4 -1 -7

The remainder of the synthetic division is -7. Therefore,

f(-6) = -7

3. Question: Use the remainder theorem and synthetic division to find f(k) for the given value of k. F(x) = 3x^4 – 17x^3 – 3x^2 + 4x + 4; k = -1/3.

Answer: We will use synthetic division with the given value of k = -1/3:

-1/3 | 3 -17 -3 4 4

| -1/3 16/9 -13/9 -1/9

|—————————-

3 -17 -1/3 3/9 31/9

The remainder of the synthetic division is 31/9. Thus,

f(-1/3) = 31/9

4. Question: Use synthetic division to decide whether the given number k is a zero of the polynomial function. If it is not, give the value of f(k). F(x) = x^2 – 8x + 15; k = 5.

Answer: To determine whether k = 5 is a zero of the polynomial function, we perform synthetic division:

5 | 1 -8 15

| 5 -15

|————-

1 -3 0

Since the remainder is 0, k = 5 is indeed a zero of the function F(x) = x^2 – 8x + 15.

5. Question: For the polynomial function, use the remainder theorem and synthetic division to find f(k). F(x) = 3x^2 + 48, k = 4i.

Answer: To find f(k), we substitute the given value of k = 4i into the polynomial function F(x). Hence,

f(4i) = 3(4i)^2 + 48

= 3(16(-1)) + 48

= -48 + 48

= 0

Thus, f(4i) = 0.

6. Question: Express f(x) in the form f(x) = (x – k)q(x) + r for the given value of k. F(x) = x^3 + 5x^2 + 7x + 2, k = -2.

Answer: To express f(x) in the desired form, we use synthetic division with k = -2:

-2 | 1 5 7 2

| -2 -6 -2

|—————-

1 3 1 0

Therefore, f(x) = (x + 2)(x^2 + 3x + 1) + 0.

Hence, f(x) = (x + 2)(x^2 + 3x + 1).