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Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason
you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete
additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight
forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for
you.
Theoretically Easier Difficulty Problem
By FTA, the number of solutions (including the possibly complex zeros and multiplicity) is equal to the degree of the
polynomial.
Consider the function . According to the Fundamental Theorem of Algebra, how many (possibly complex-valued) zeros are there
(counting multiplicity) for ?
Theoretically Medium Difficulty Problem
Consider the function .
Remember that the leading term is the entire term that has the highest degree in the
polynomial.
You may have to simplify (combine like terms) the polynomial before you have the entire term.
What is the leading term in this polynomial?
Remember that the leading coefficient is the constant that is multiplying the entire leading term.
You may have to simplify (combine like terms) the polynomial before you have the entire term.
What is the leading coefficient in this polynomial?
You should use the Fundamental Theorem of Algebra (FTA) to get this.
By FTA, the number of solutions (including the possibly complex zeros and multiplicity) is equal to the degree of the
polynomial.
How many (possibly complex-valued) zeros are there (counting multiplicity) for ?
Theoretically Harder Difficulty Problem
You should use the Fundamental Theorem of Algebra (FTA) to get this.
By FTA, the number of solutions (including the possibly complex zeros and multiplicity) is equal to the degree of the
polynomial.
Consider the function . According to the Fundamental Theorem of Algebra, how many (possibly complex-valued) zeros are there
(counting multiplicity) for ?
You know how many total zeros the polynomial might have, so if we are only interested in a subset of that total, the number
from before must be an upper bound!
By FTA, the number of solutions (including the possibly complex zeros and multiplicity) is equal to the degree of the
polynomial.
There are at leastexactlyat most real-valued solutions.
If we know that the value is an upper bound, can we have fewer than that number?
This means there could bemore thanexactlyless than real-valued zeros.
Precalculus Algebra