Consider the function f(x)=x^3+4x^2+kx-4. The remainder from f(x)/(x+2) is twice the remainder from f(x) / (x-2). Determine the value of k

first, sub in both x values

first (x-2), x=2
f(2) = (2)^3 + 4(2)^2 + k(2) - 4
f(2) = 20 + 2k

same with (x+2), x= -2
f(-2) = (-2)^3 + 4(-2)^2 + k(-2) - 4
f(-2) = 4 - 2k

we know that f(2) (x - 2) remainder is 2 times larger than f(-2) (x + 2) remainder, so:

2(20 + 2k) = 4 - 2k
40 + 4k = 4 - 2k
36 = -6k
therefore, k = -6

Well, well, solving a math problem with a touch of humor, here we go!

Let's denote the remainder from f(x) divided by (x+2) as R1 and the remainder from f(x) divided by (x-2) as R2.

According to the given information, we have R1 = 2R2.

Now, to find the remainder from f(x) divided by (x+2), we can simply substitute x = -2 into the equation:

f(-2) = (-2)^3 + 4(-2)^2 + k(-2) - 4

Now, let's find the remainder from f(x) divided by (x-2) by substituting x = 2:

f(2) = 2^3 + 4(2)^2 + k(2) - 4

Since R1 = 2R2, we have:

(-2)^3 + 4(-2)^2 + k(-2) - 4 = 2^3 + 4(2)^2 + k(2) - 4

Now it's all about simplifying and solving for k:

-8 + 4(4) - 2k - 4 = 8 + 4(4) + 2k - 4

-8 + 16 - 2k - 4 = 8 + 16 + 2k - 4

4 - 2k - 4 = 24 + 2k - 4

-2k = 24 + 2k

Combine like terms:

-4k = 24

Finally, divide both sides by -4:

k = -6

So the value of k that satisfies the given condition is -6!

Hope that brought a smile to your face while solving the math problem!

To find the remainder when the polynomial f(x) is divided by (x+2), we can use the Remainder Theorem.

The Remainder Theorem states that if you divide a polynomial f(x) by (x-a), the remainder will be equal to f(a).

So, when f(x) is divided by (x+2), the remainder is f(-2).
Therefore, the remainder from f(x) / (x+2) is f(-2).

Similarly, when f(x) is divided by (x-2), the remainder is f(2).
Therefore, the remainder from f(x) / (x-2) is f(2).

The problem states that the remainder from f(x) / (x+2) is twice the remainder from f(x) / (x-2).

In equation form, we have:
f(-2) = 2 * f(2)

Using the function f(x) = x^3 + 4x^2 + kx - 4, we can substitute the values of -2 and 2 to solve for k:

f(-2) = (-2)^3 + 4(-2)^2 + k(-2) - 4
f(2) = (2)^3 + 4(2)^2 + k(2) - 4

Simplifying these equations, we get:
f(-2) = -8 + 16 - 2k - 4
f(2) = 8 + 16 + 2k - 4

Now we can substitute these values back into our original equation:

-8 + 16 - 2k - 4 = 2(8 + 16 + 2k - 4)

Simplifying this equation:

4 - 2k = 32 + 32 + 4k - 8

Combine like terms:

-2k - 4k = 32 + 32 - 4 + 8 - 4

-6k = 64

Dividing both sides by -6:

k = -64/6

Simplifying the fraction gives:

k = -32/3

Therefore, the value of k is -32/3.

To find the value of k, we need to use a property of remainders when dividing polynomials.

When we divide a polynomial f(x) by (x - a), the remainder is given by substituting a into the polynomial f(x). In other words, f(a) is equal to the remainder when f(x) is divided by (x - a).

Thus, the remainder from f(x) divided by (x + 2) is given by f(-2), and the remainder from f(x) divided by (x - 2) is given by f(2).

According to the given information, the remainder from f(x) divided by (x + 2) is twice the remainder from f(x) divided by (x - 2). Mathematically, we can write this as:

f(-2) = 2 * f(2)

Let's find f(-2) first:

f(-2) = (-2)^3 + 4(-2)^2 + k(-2) - 4 = -8 + 16 - 2k - 4 = 4 - 2k

Next, find f(2):

f(2) = (2)^3 + 4(2)^2 + k(2) - 4 = 8 + 16 + 2k - 4 = 20 + 2k

Now, equate the two expressions (since one is twice the other):

4 - 2k = 2(20 + 2k)

Simplify the equation:

4 - 2k = 40 + 4k

Add 2k to both sides:

4 = 40 + 6k

Subtract 40 from both sides:

-36 = 6k

Divide both sides by 6 to solve for k:

k = -6

Therefore, the value of k is -6.

A little synthetic division yields:

f(x) = (x+2)(x^2+2x+k-4) + (-4-2k)
f(x) = (x-2)(x^2+6x+k+12) + (2k+20)

-4-2k = 2k+20
k = -6