Express the function in the form
f(x) = (x-k)q(x) + r for the given value of k.
f(x) = 15x^4 + 10x^3 - 6x^2 + 14
Value of k = -2/3
I do not understand what I am suppose to do!
For the given f(x), you will need to use synthetic to find the quotient q(x) and the remainder, r.
Since you already figured out how to do synthetic division, you should have no problem tackling this one.
Post your answer for a check if you wish.
i did -2/3 divided by 15, 10, -6, 0 and 14
I got 15, 0, -6 , 4 ,11.4
Is this correct?
Yes, it is correct. A clearer way to write your answer is to put it in polynomial form:
q(x)=15x^3-6x+4
r=34/3=11 1/3
Note that the remainder is 11 1/3 and not 11.4.
Thank You
You're very welcome!
To express the function in the given form, f(x) = (x-k)q(x) + r, we need to divide the given polynomial by (x - k), where k is the given value. In this case, the given value of k is -2/3.
To perform the division, we can use polynomial long division or synthetic division. Here, we'll use synthetic division for simplicity.
Step 1: Set up the synthetic division table:
-2/3 | 15 10 -6 0 14
Step 2: Bring down the coefficient of the highest degree term (15):
-2/3 | 15 10 -6 0 14
|
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Step 3: Multiply the divisor (-2/3) by each term in the row above it and write the result below:
-2/3 | 15 10 -6 0 14
| -10/3 -5 11 -22
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Step 4: Add the corresponding terms:
-2/3 | 15 10 -6 0 14
| -10/3 -5 11 -22
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15 0 -11 11 -8
The last value in the bottom row (-8) represents the remainder (r) when f(x) is divided by (x - k). Therefore, r = -8.
The remaining values in the bottom row represent the coefficients of the quotient polynomial, q(x). Since the original polynomial is of degree 4, the quotient polynomial will be of degree 3.
Therefore, the expression for f(x) in the requested form is:
f(x) = (x - (-2/3))(15x^3 + 0x^2 - 11x + 11) + (-8)
Simplifying this equation gives:
f(x) = (x + 2/3)(15x^3 - 11x + 11) - 8
So, the function f(x) expressed in the required form with the given value of k = -2/3 is:
f(x) = (x + 2/3)(15x^3 - 11x + 11) - 8.