Write the function in the form
f(x) = (x − k)q(x) + r
for the given value of k.
f(x) = −4x^3 + 6x^2 + 8x + 2, k = 1 − _/2
(square root of 2 if you didn't get what I meant I think _/ looks a bit like the square root symbol so I used it to represent it)
since the coefficients are rational, any irrational roots come in conjugate pairs, so another root is 1+√2
(x-(1-√2))(x-(1+√2)) = x^2-2x-1
So, dividing by that we have
(−4x^3 + 6x^2 + 8x + 2)/(x^2-2x-1) = -4x-2 with no remainder
So,
f(x) = (-4x-2)(x-(1+√2))(x-(1-√2)) = (x-(1-√2))(-4x^2+(2+4√2)x+2+2√2) + 0
To write the function in the desired form, we need to find the quotient and remainder when dividing f(x) by (x - k). In this case, k is 1 - √2.
Step 1: Divide the polynomial f(x) by (x - k).
- First, let's convert the value of k = 1 - √2 to decimal form for convenience. Solving 1 - √2 ≈ -0.4142.
- Now, perform long division to divide the polynomial f(x) = -4x^3 + 6x^2 + 8x + 2 by (x - k):
______________________
x - k ) -4x^3 + 6x^2 + 8x + 2
- (-4x^3 + 0x^2 + 4√2x^2 + 6√2x - 4√2x - 6√2 + 8x + 2)
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0x^2 + (8 - 4√2)x - (6√2 - 6√2)
- Simplifying the above, we get:
0x^2 + (8 - 4√2)x - (0)
Step 2: Rewrite the division equation as the desired form.
In the desired form, we have:
f(x) = (x - k)q(x) + r
- From the long division, we can see that the quotient, q(x), is 0, and the remainder, r, is 0x^2 + (8 - 4√2)x - (0).
Therefore, the function can be written in the desired form as:
f(x) = (x - (1 - √2))(0) + (0x^2 + (8 - 4√2)x - 0)
Simplifying further, we have:
f(x) = 0x^2 + (8 - 4√2)x
Thus, the function is f(x) = 0x^2 + (8 - 4√2)x.