Applying properties of operations on complex numbers to solve problems

Which quadratic has COMPLEX roots?

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Determine the linear factor according to the given graph. Assume the function has degree 2.

• (x - 1)(x + 2)

• -(x - 1)(x + 2)

• (x + 1)(x - 2)

• -(x + 1)(x - 2)

Which root has even multiplicity?

• 1
• -4
• -1
• 2

Write in factored form using the following roots.

7, -2i

• (x-7)(x-2i)

• (x-7)(x+2i)(x-2i)

• (x-7)(x+2i)

• (x-7)(x+7)(x-2i)

f(x)=(x+4)(x-3)(x-2)
List the zeros for this function.

• x=-4, x=3, x=-2
• x=-4, x=3, x=2
• x=-4, x=3, x=2
• x=4, x=3, x=2

One solution to the equation  $x^2+px+q=0$   is x = 1 + 4i. You determine the factored form to be 

 $(X-(1+4i))(X-(1-4i))\ =0$  
Use this to determine the standard form.

•  $x^2-2x+1-16i^2=0$  
  

•  $x^2-2x+17=0$  
  

• 17

•  $x^2+2x+1+8xi+4i-16i^2=0$  
  

Answer explanation

How many REAL solutions does this quadratic have?

• Infinitely many

• One

• Two

• None

• It can't be done.

Answer explanation

The Factor Theorem states if (x-a) is a factor of f(x), then f(a)=0.

Likewise, if f(a)=0, then (x-a) is a factor of f(x).

The Fundamental Theorem of Algebra tell us two things. Which of the options does not come from the Fundamental Theorem of Algebra?

• A nth degree polynomial has at least 1 complex solution

• An nth degree polynomial always has 2 x-intercepts

• A nth degree polynomial has n solutions (roots).

Match the equation to the graph.

• y=-(x+1)(x-2)(x-4)
• y=(x+1)²(x-3)(x-1)³
• y = 3x³ – 2x + 1
• y=(x+1)³(x-3)(x-1)²