Answer:
{ -0.5 - √2, 0, -.5 + √2 }
Explanation:
Let's rewrite these two functions in a vertical column:
f(x)=9 x^{3} +2x^2-5x+4
-g(x)=5x^3-7x+4
Now combine (through subtraction) like terms, in order of descending powers of x:
f(x) - g(x) = 4x^ 3 + 2x^2 + 2x
Let's do some preliminary factoring here. Factor 2x out of each term, obtaining
f(x) - g(x) = 2x(2x^2 + x + 1)
First setting 2x = 0, we find that x = 0 is one solution.
Applying the quadratic formula to 2x^2 + x + 1, we see that a = 2, b = 1 and c = 1. Then the two roots of this quadratic are:
-1 plus or minus √(1^2 - 4(2)(1) ) -1 ± √(8)
x = ----------------------------------------------- = ---------------
2(1) 2
This simplifies to:
-1 plus or minus √(1^2 - 4(2)(1) ) -1 ± 2√(2)
x = ----------------------------------------------- = ---------------
2(1) 2
Thus, the roots of f(x) - g(x) are { -0.5 - √2, 0, -.5 + √2 }