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Wize High School Algebra II Textbook (Common Core) > Polynomial Functions

Fundamental Theorem of Algebra

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Fundamental Theorem of Algebra


When working to factor various polynomials a great question comes up: "How do we know the polynomial actually breaks into factors?" Fortunately this is answered by the Fundamental Theorem of Algebra

What it says

According to the fundamental theorem of algebra every polynomial with complex coefficients will have at least one complex root, as long as its not a constant function.

In plain English this means that many of the polynomials we work with can be broken down into factors, as long as we include complex numbers as part of our factors.

Example
Which of the functions below does the Fundamental Theorem of Algebra apply to?
y=4x2−17x+1y = 4x^2-17x + 1
ANSWER: The theorem applies since this is a polynomial

y=x2+4y = x^2 + 4
ANSWER: The theorem applies since this is a polynomial. This one happens to factor into (x−2i)(x+2i)(x - 2i)(x+2i)

y=3x1/2−x+1y = 3x^{1/2} - x + 1
ANSWER: This is not a polynomial. The theorem can't be applied, and so gives us no information.

y=7y = 7
ANSWER: This is a polynomial, but it is a constant function. This means the theorem does not apply.

Watch Out!
The Fundamental Theorem of Algebra says it is possible to break down a polynomial into factors if we include complex number. Unfortunately it does not tell us exactly how to find those factors. To find the factors we must use a whole host of other techniques.

Practice with the Fundamental Theorem of Algebra


Determine if the Fundamental Theorem of Algebra applies to the function or not. Remember the theorem helps us determine if it might be factored.

1. f(x)=21+xf(x) = \displaystyle\frac{2}{1+x}

2. g(x)=13x4−2x2+2g(x) = \displaystyle\frac{1}{3}x^4-2x^2+\sqrt{2}

3. h(x)=x6+4x3+4h(x) = x^6+4x^3 + 4
1. f(x)=21+xf(x) = \displaystyle\frac{2}{1+x}

2. g(x)=13x4−2x2+2g(x) = \displaystyle\frac{1}{3}x^4-2x^2+\sqrt{2}

3. h(x)=x6+4x3+4h(x) = x^6+4x^3 + 4