You are about to erase your work on this activity. Are you sure you want to do this?

Updated Version Available

There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?

Mathematical Expression Editor

1 : Consider the function . According to the Fundamental Theorem of Algebra,
how many (possibly complex-valued) zeros are there for ?

The Fundamental
Theorem of Algebra says that the number of zeros is exactly equal to the degree of
the polynomial.

1.1 : There are at leastexactlyat most real-valued solutions.

Although we know there are exactly the same number of
solutions as the degree of the polynomial, some of them might be complex-valued. So
we only know that at most there are real-valued solutions (since they may all
be real) but some could be complex, so we don’t know exactly how many
real-valued solutions there are; at least not without doing a bunch more work.

1.1.1 : This means there could bemore thanexactlyless than real-valued zeros.

Remember that this means there definitely could be a lower
number of real valued solutions than complex valued solutions. In particular, if
there are irreducible quadratic factors; but we will cover this more later!

1.2 : What is the leading term in this polynomial?

Remember that you may
need to simplify (combine like terms) the polynomial to get the correct leading term.

1.3 : What is the leading coefficient in this polynomial?

Remember that you
may need to simplify (combine like terms) the polynomial to get the correct leading
coefficient.

2 : Consider the function . According to the Fundamental Theorem of Algebra,
how many (possibly complex-valued) zeros are there for ?

The Fundamental
Theorem of Algebra says that the number of zeros is exactly equal to the degree of
the polynomial.

2.1 : There are at leastexactlyat most real-valued solutions.

Although we know there are exactly the same number of
solutions as the degree of the polynomial, some of them might be complex-valued. So
we only know that at most there are real-valued solutions (since they may all
be real) but some could be complex, so we don’t know exactly how many
real-valued solutions there are; at least not without doing a bunch more work.

2.1.1 : This means there could bemore thanexactlyless than real-valued zeros.

Remember that this means there definitely could be a lower
number of real valued solutions than complex valued solutions. In particular, if
there are irreducible quadratic factors; but we will cover this more later!

2.2 : What is the leading term in this polynomial?

Remember that you may
need to simplify (combine like terms) the polynomial to get the correct leading term.

2.3 : What is the leading coefficient in this polynomial?

Remember that you
may need to simplify (combine like terms) the polynomial to get the correct leading
coefficient.

3 : Consider the function . According to the Fundamental Theorem of Algebra,
how many (possibly complex-valued) zeros are there for ?

The Fundamental
Theorem of Algebra says that the number of zeros is exactly equal to the degree of
the polynomial.

3.1 : There are at leastexactlyat most real-valued solutions.

Although we know there are exactly the same number of
solutions as the degree of the polynomial, some of them might be complex-valued. So
we only know that at most there are real-valued solutions (since they may all
be real) but some could be complex, so we don’t know exactly how many
real-valued solutions there are; at least not without doing a bunch more work.

3.1.1 : This means there could bemore thanexactlyless than real-valued zeros.

Remember that this means there definitely could be a lower
number of real valued solutions than complex valued solutions. In particular, if
there are irreducible quadratic factors; but we will cover this more later!

3.2 : What is the leading term in this polynomial?

Remember that you may
need to simplify (combine like terms) the polynomial to get the correct leading term.

3.3 : What is the leading coefficient in this polynomial?

Remember that you
may need to simplify (combine like terms) the polynomial to get the correct leading
coefficient.