# 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?

- 6
- 3
- 7
- 1

2.

How many roots does the polynomial [math]x^2-6-x^4-x^6[/math] have?

- 6
- 5
- 2
- 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?

- 6
- 0
- 5
- 1

4.

What is the minimum number of real roots that the polynomial [math]x^2-6-x^4-x^6[/math] can have?

- 6
- 0
- 5
- 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?

- 7
- 1
- 3
- 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?

- 5
- 4
- 2
- 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.)

- 5
- 4
- 3
- 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?

- Yes, this is a known exception.
- No, this is not a polynomial, it is a quadratic function.
- No, if the quadratic formula is used, the other root is found.
- 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?

- Quadratic polynomials always intersect the x-axis.
- 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.
- When a quadratic polynomial doesn't cross the x-axis, this simply implies that its roots are complex with non-zero imaginary parts.
- 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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