4db8d1…: “With an important gulf”

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1 : Consider the function $f(x) = \sage {p1f1} + \sage {p1f2} + \sage {p1f3} + \sage {p1f4}$ . According to the Fundamental Theorem of Algebra, how many (possibly complex-valued) zeros are there for $f(x)$ ? $\answer {\sage {p1ans1}}$

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 least exactly at most $\answer {\sage {p1ans1}}$ 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 $\sage {p1ans1}$ 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 be more than exactly less than $\sage {p1ans1}$ 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? $\answer {\sage {p1ans2}}$

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? $\answer {\sage {p1ans3}}$

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

2 : Consider the function $f(x) = \sage {p2f1} + \sage {p2f2} + \sage {p2f3} + \sage {p2f4}$ . According to the Fundamental Theorem of Algebra, how many (possibly complex-valued) zeros are there for $f(x)$ ? $\answer {\sage {p2ans1}}$

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 least exactly at most $\answer {\sage {p2ans1}}$ 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 $\sage {p2ans1}$ 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 be more than exactly less than $\sage {p2ans1}$ real-valued zeros.

2.2 : What is the leading term in this polynomial? $\answer {\sage {p2ans2}}$

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? $\answer {\sage {p2ans3}}$

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

3 : Consider the function $f(x) = \sage {p3f1} + \sage {p3f2} + \sage {p3f3} + \sage {p3f4}$ . According to the Fundamental Theorem of Algebra, how many (possibly complex-valued) zeros are there for $f(x)$ ? $\answer {\sage {p3ans1}}$

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 least exactly at most $\answer {\sage {p3ans1}}$ 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 $\sage {p3ans1}$ 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 be more than exactly less than $\sage {p3ans1}$ real-valued zeros.

3.2 : What is the leading term in this polynomial? $\answer {\sage {p3ans2}}$

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? $\answer {\sage {p3ans3}}$

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