Question

F(x)=9 x^{3} +2x^2-5x+4 and g(x)=5x^3-7x+4. What is f(x)-g(x)? Show all of your steps and write your finial answer in factored form.

Answer 1

Answer:

{ -0.5 - √2,  0,  -.5 + √2 }

Step-by-step 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 }