Question

The graph of F(x) shown below resembles the graph of G(x)= x4, but it has

been vertically stretched. Which of the following could be the equation of F(X)
?

Answer 1

• x = 0.

Equation:

• 
  `g*(x)-(x^4)=0`

Explanation:

Step 1:

• Pull out like factors:
• 
  `gx - x^4 = x * (g - x^3)`

Trying to factor as a Difference of Cubes:

• Factoring:
  `g - x^3`
• Theory : A difference of two perfect cubes, a^3 - b^3 can be factored into
• (a-b) • (a^2 +ab +b^2)
• Proof : (a-b)•(a^2+ab+b^2) =
• a^3+a^2b+ab^2-ba^2-b^2a-b^3 =
• a^3+(a^2b-ba^2)+(ab^2-b^2a)-b^3 =
• a^3+0+0-b^3 =
• a^3-b^3
• Check : g^1 is not a cube !!
• Ruling : Binomial cannot be factored as the difference of two perfect cubes

Equation at end of step 1:

• 
  `x * (g - x^3) = 0`

Step 2:

• A product of several terms equals zero.
• When a product of two or more terms equals zero, then at least one of the terms must be zero.
• We shall now solve each term = 0 separately
• In other words, we are going to solve as many equations as there are terms in the product
• Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

• Solve
  `g - x^3 = 0`
• In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.
• We shall not handle this type of equations at this time.

Solution:

• x=0.