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Example Question #1 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
What are the roots of
including complex roots, if they exist?
One of the roots is because if we plug in 1, we get 0. We can factor the polynomial as
So now we solve the roots of .
The root will not be real.
The roots of this polynomial are .
So, the roots are
Example Question #2 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
The polynomial has a real zero at 1.5. Find the other two zeros.
If this polynomial has a real zero at 1.5, that means that the polynomial has a factor that when set equal to zero has a solution of . We can figure out what this is this way:
multiply both sides by 2
is the factor
Now that we have one factor, we can divide to find the other two solutions:
To finish solving, we can use the quadratic formula with the resulting quadratic, :
Example Question #3 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
If the real zero of the polynomial is 3, what are the complex zeros?
We know that the real zero of this polynomial is 3, so one of the factors must be . To find the other factors, we can divide the original polynomial by
, either by long division or synthetic division:
This gives us a second factor of which we can solve using the quadratic formula:
Example Question #4 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
The polynomial intersects the x-axis at point
. Find the other two solutions.
Since we know that one of the zeros of this polynomial is 3, we know that one of the factors is . To find the other two zeros, we can divide the original polynomial by
, either with long division or with synthetic division:
This gives us the second factor of . We can get our solutions by using the quadratic formula:
Example Question #5 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
Find all the real and complex zeroes of the following equation:
First, factorize the equation using grouping of common terms:
Next, setting each expression in parenthesis equal to zero yields the answers.
Example Question #6 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
Find all the zeroes of the following equation and their multiplicity:
(multiplicity of 2 on 0, multiplicity of 1 on
(multiplicity of 1 on 0, multiplicity of 2 on
(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
First, pull out the common t and then factorize using quadratic factoring rules:
This equation has solutions at 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 #6 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
Find a fourth degree polynomial whose zeroes are -2, 5, and
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 #7 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra
The third degree polynomial expression has a real zero at
. Find all of the complex zeroes.
First, factor the expression by grouping:
To find the complex zeroes, set the term equal to zero:
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:
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:
(multiplicity of 1 on 0, multiplicity of 2 on
)
(multiplicity of 2 on 0, multiplicity of 1 on
)
(multiplicity of 1 on 0, multiplicity of 2 on
)
(multiplicity of 2 on 0, multiplicity of 1 on
)
(multiplicity of 1 on 0, multiplicity of 2 on
)
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.
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