Answer:
p(x) = -4*x^4 - 2*x^3 + 3*x^2 + 1*x - 2
Explanation:
We want to find a polynomial p(x), such that if we add that polynomial to:
f(x) = 4*x^4 + 2*x^3 - 2*x^2 + x - 1
we get:
g(x) = x^2 + 2*x - 3
This is:
f(x) + p(x) = g(x)
Notice that f(x) is a polynomial of degree 4 and g(x) is a polinomial of degree 2, so p(x) must be also a polynomial of degree 4.
p(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e
Then we get:
(4*x^4 + 2*x^3 - 2*x^2 + x - 1) + (a*x^4 + b*x^3 + c*x^2 + d*x + e) = x^2 + 2*x - 3
We can simplify the left side to:
(4 + a)*x^4 + (2 + b)*x^3 + (-2 + c)*x^2 + (1 + d)*x + (-1 + e) = x^2 + 2*x - 3
Because in the right side we do not have terms with exponent 4 and 3, we must have that:
4 + a = 0
2 + b = 0
and for the other exponents of x we just match the exponent in the left side with the correspondent one in the right side:
(-2 + c) = 1
(1 + d) = 2
(-1 + e) = -3
Solving the 5 equations we get:
a = -4
b = -2
c = 1 + 2 = 3
d = 2 - 1 = 1
e = -3 + 1 = -2
Then the equation for p(x) is:
p(x) = -4*x^4 - 2*x^3 + 3*x^2 + 1*x - 2