Precalculus : Fundamental Theorem of Algebra

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #9 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find all the real and complex zeros of the following equation: 

Possible Answers:

Correct answer:

Explanation:

First, factorize the equation using grouping of common terms:

Next, setting each expression in parentheses equal to zero yields the answers.

Example Question #10 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find all the zeroes of the following equation and their multiplicity: 

Possible Answers:

 (multiplicity of 2 on 0, multiplicity of 1 on )

 (multiplicity of 1 on 0, multiplicity of 2 on )

 (multiplicity of 1 on 0, multiplicity of 2 on )

 (multiplicity of 2 on 0, multiplicity of 1 on )

Correct answer:

 (multiplicity of 1 on 0, multiplicity of 2 on )

Explanation:

First, pull out the common t and then factorize using quadratic factoring rules: 

This equation has a solution as two values: when , and when . Therefore, But since the degree on the former equation is one and the degree on the latter equation is two, the multiplicities are 1 and 2 respectively.

Example Question #21 : Fundamental Theorem Of Algebra

Find a fourth-degree polynomial whose zeroes are , and 

Possible Answers:

Correct answer:

Explanation:

This one is a bit of a journey. The expressions for the first two zeroes are easily calculated,  and  respectively. The last expression must be broken up into two equations:  which are then set equal to zero to yield the expressions  and

Finally, we multiply together all of the parenthesized expressions, which multiplies out to 

Example Question #22 : Fundamental Theorem Of Algebra

The third-degree polynomial expression  has a real zero at . Find all of the complex zeroes.

Possible Answers:

Correct answer:

Explanation:

First, factor the expression by grouping:

To find the complex zeroes, set the term  equal to zero: 

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