Answer:
f(x) = (x + 1)*(4*x² - 3*x + 4) - 12
Explanation:
We have:
f(x) = 4*x³ + x² + x - 8
We want to write this in:
f(x) = (x - k)*q(x) + r.
with k = -1
Then we want to write:
4*x³ + x² + x - 8 = (x - (-1))*q(x) + r
4*x³ + x² + x - 8 = (x + 1)*q(x) + r
Because f(x) is polynomial of degree 3, we know that q(x) must be a polynomial of degree 2.
then:
q(x) = a*x² + b*x + c
Then:
4*x³ + x² + x - 8 = (x + 1)*(a*x² + b*x + c) + r
4*x³ + x² + x - 8 = a*x³ + b*x² + c*x + a*x² + b*x + c + r
if we take common factors in the right side we get:
4*x³ + x² + x - 8 = a*x³ + (b + a)*x² + (c + b)*x + (c + r)
Now, we must have:
4*x³ = a*x³
then:
4 = a
We also must have:
x² = (b + a)*x²
1 = (b + 4)
1 - 4 = b
-3 = b
We also must have:
x = (c + b)*x
1 = (c + (-3))
1 + 3 = c
4 = c
And finally:
- 8 = (c + r)
-8 = 4 + r
-8 - 4 = r
-12 = r
Then:
f(x) = (x + 1)*(4*x² - 3*x + 4) - 12