A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?
3 meters by 4 meters by 7 meters
Access these online resources for additional instruction and practice with zeros of polynomial functions.
To find
determine the remainder of the polynomial
when it is divided by
See
[link] .
is a zero of
if and only if
is a factor of
See
[link] .
Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See
[link] and
[link] .
When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
Synthetic division can be used to find the zeros of a polynomial function. See
[link] .
According to the Fundamental Theorem, every polynomial function has at least one complex zero. See
[link] .
Every polynomial function with degree greater than 0 has at least one complex zero.
Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form
where
is a complex number. See
[link] .
The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
The number of negative real zeros of a polynomial function is either the number of sign changes of
or less than the number of sign changes by an even integer. See
[link] .
Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See
[link] .
Section exercises
Verbal
Describe a use for the Remainder Theorem.
The theorem can be used to evaluate a polynomial.
Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.
What is the difference between rational and real zeros?
Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.
If Descartes’ Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?
If synthetic division reveals a zero, why should we try that value again as a possible solution?
Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.
Algebraic
For the following exercises, use the Remainder Theorem to find the remainder.
For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.
