Fundamental Theorem of Algebra (Grades 11-12)

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Fundamental Theorem of Algebra

1. 
How many roots does the polynomial [math]x^3-x^6-x-x^7[/math] have?
  1. 6
  2. 3
  3. 7
  4. 1
2. 
How many roots does the polynomial [math]x^2-6-x^4-x^6[/math] have?
  1. 6
  2. 5
  3. 2
  4. 1
3. 
What is the maximum number of purely imaginary roots that the polynomial [math]x^2-6-x^4-x^6[/math] can have?
  1. 6
  2. 0
  3. 5
  4. 1
4. 
What is the minimum number of real roots that the polynomial [math]x^2-6-x^4-x^6[/math] can have?
  1. 6
  2. 0
  3. 5
  4. 1
5. 
What is the maximum number of purely imaginary roots that the polynomial [math]x^3-x^6-x-x^7[/math] can have?
  1. 7
  2. 1
  3. 3
  4. 6
6. 
What is the maximum number of complex roots with non-zero imaginary part, [math]a+bi, b!=0[/math], the polynomial [math]4x^5 - 5x + 6[/math] can have?
  1. 5
  2. 4
  3. 2
  4. 0
7. 
How many complex roots does the polynomial [math](x-2)^3(x^2+1)[/math] have? (Remember, real numbers are a subset of complex numbers.)
  1. 5
  2. 4
  3. 3
  4. 2
8. 
Jeremy is working with the Fundamental Theorem of Algebra, and thinks he's found an exception. Looking at [math]f(x) = 4(x-1)^2[/math], this will result in only one root, [math]x=1[/math]. Therefore, despite this being a second degree polynomial, there is only one root. Is this correct?
  1. Yes, this is a known exception.
  2. No, this is not a polynomial, it is a quadratic function.
  3. No, if the quadratic formula is used, the other root is found.
  4. No, this root has multiplicity of 2, which means it counts as two roots.
9. 
Looking at the graph of a quadratic polynomial, roots or zeros correspond to where the graph crosses the x-axis. When the graph just touches the x-axis, this corresponds to a double root. The Fundamental Theorem of Algebra states that a quadratic polynomial will always have 2 roots. How is this reconciled with a quadratic polynomial whose graph does not intersect the x-axis?
  1. Quadratic polynomials always intersect the x-axis.
  2. If a quadratic polynomial doesn't cross the x-axis it is no longer a polynomial, and the Fundamental Theorem of Algebra no longer applies.
  3. When a quadratic polynomial doesn't cross the x-axis, this simply implies that its roots are complex with non-zero imaginary parts.
  4. Simply translate the quadratic polynomial till it does cross the x-axis.
10. 
The Fundamental Theorem of Algebra is sometimes stated as "A non-constant, single-variable polynomial with complex coefficients has at least one complex root". Another form of the theorem states "A single-variable polynomial of degree n has n roots" (taking into account multiplicity if necessary). Show that the first statement leads to the second statement for a quadratic polynomial (degree 2).








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