A new bakery offers decorated sheet cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided bythe remainder may be found quickly by evaluating the polynomial function atthat is,Let’s walk through the proof of the theorem.
Recall that the Division Algorithm states that, given a polynomial dividendand a non-zero polynomial divisorwhere the degree ofis less than or equal to the degree of, there exist unique polynomialsandsuch that
If the divisor,isthis takes the form
Since the divisoris linear, the remainder will be a constant,And, if we evaluate this forwe have
In other words,is the remainder obtained by dividingby
If a polynomialis divided bythen the remainder is the value
Given a polynomial functionevaluateatusing the Remainder Theorem.
Use the Remainder Theorem to evaluate at
To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by
The remainder is 25. Therefore,
We can check our answer by evaluating
Use the Remainder Theorem to evaluate at
The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.
If is a zero, then the remainder is and or
Notice, written in this form,is a factor ofWe can conclude if is a zero ofthenis a factor of
Similarly, if is a factor of then the remainder of the Division Algorithm is 0. This tells us that is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree in the complex number system will have zeros. We can use the Factor Theorem to completely factor a polynomial into the product of factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.
According to the Factor Theorem, is a zero of if and only if is a factor of
Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.
Show that is a factor ofFind the remaining factors. Use the factors to determine the zeros of the polynomial.
We can use synthetic division to show thatis a factor of the polynomial.
The remainder is zero, sois a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:
We can factor the quadratic factor to write the polynomial as
By the Factor Theorem, the zeros ofare –2, 3, and 5.
Use the Factor Theorem to find the zeros of given that is a factor of the polynomial.
The zeros are 2, –2, and –4.
Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial
Consider a quadratic function with two zeros, and By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.
Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.
We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.
The Rational Zero Theorem states that, if the polynomialhas integer coefficients, then every rational zero of has the formwhereis a factor of the constant termandis a factor of the leading coefficient
When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
Given a polynomial functionuse the Rational Zero Theorem to find rational zeros.
List all possible rational zeros of
The only possible rational zeros of are the quotients of the factors of the last term, –4, and the factors of the leading coefficient, 2.
The constant term is –4; the factors of –4 are
The leading coefficient is 2; the factors of 2 are
If any of the four real zeros are rational zeros, then they will be of one of the following factors of –4 divided by one of the factors of 2.
Note that andwhich have already been listed. So we can shorten our list.
Use the Rational Zero Theorem to find the rational zeros of
The Rational Zero Theorem tells us that ifis a zero of thenis a factor of 1 andis a factor of 2.
The factors of 1 areand the factors of 2 areandThe possible values forareandThese are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values forin
Of those, are not zeros of1 is the only rational zero of
Use the Rational Zero Theorem to find the rational zeros of
There are no rational zeros.
The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.
Given a polynomial functionuse synthetic division to find its zeros.
Find the zeros of
The Rational Zero Theorem tells us that ifis a zero ofthen is a factor of –1 andis a factor of 4.
The factors of are and the factors of areandThe possible values forareand These are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let’s begin with 1.
Dividing by gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as
The quadratic is a perfect square. can be written as
We already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the other zero, we can set the factor equal to 0.
The zeros of the function are 1 andwith multiplicity 2.
Look at the graph of the functionin [link]. Notice, atthe graph bounces off the x-axis, indicating the even multiplicity (2,4,6…) for the zero Atthe graph crosses the x-axis, indicating the odd multiplicity (1,3,5…) for the zero
Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.
Supposeis a polynomial function of degree four, andThe Fundamental Theorem of Algebra states that there is at least one complex solution, call itBy the Factor Theorem, we can writeas a product ofand a polynomial quotient. Sinceis linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call itSo we can write the polynomial quotient as a product ofand a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of
The Fundamental Theorem of Algebra states that, if f(x) is a polynomial of degree n > 0, then f(x) has at least one complex zero.
We can use this theorem to argue that, ifis a polynomial of degreeand is a non-zero real number, thenhas exactly linear factors
whereare complex numbers. Therefore,hasroots if we allow for multiplicities.
Does every polynomial have at least one imaginary zero?
No. Real numbers are a subset of complex numbers, but not the other way around. A complex number is not necessarily imaginary. Real numbers are also complex numbers.
Find the zeros of
The Rational Zero Theorem tells us that ifis a zero ofthenis a factor of 3 andis a factor of 3.
The factors of 3 areand The possible values forand therefore the possible rational zeros for the function, are We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let’s begin with –3.
Dividing bygives a remainder of 0, so –3 is a zero of the function. The polynomial can be written as
We can then set the quadratic equal to 0 and solve to find the other zeros of the function.
The zeros of are –3 and
Look at the graph of the functionin [link]. Notice that, atthe graph crosses the x-axis, indicating an odd multiplicity (1) for the zeroAlso note the presence of the two turning points. This means that, since there is a 3rd degree polynomial, we are looking at the maximum number of turning points. So, the end behavior of increasing without bound to the right and decreasing without bound to the left will continue. Thus, all the x-intercepts for the function are shown. So either the multiplicity ofis 1 and there are two complex solutions, which is what we found, or the multiplicity atis three. Either way, our result is correct.
Find the zeros of
The zeros are
A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree will havezeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into factors. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the formwhereis a complex number.
Let be a polynomial function with real coefficients, and suppose is a zero of Then, by the Factor Theorem, is a factor of For to have real coefficients, must also be a factor of This is true because any factor other than when multiplied by will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function with real coefficients has a complex zero then the complex conjugate must also be a zero ofThis is called the Complex Conjugate Theorem.
According to the Linear Factorization Theorem, a polynomial function will have the same number of factors as its degree, and each factor will be in the form, where is a complex number.
If the polynomial function has real coefficients and a complex zero in the form then the complex conjugate of the zero, is also a zero.
Given the zeros of a polynomial functionand a point (c, f(c)) on the graph ofuse the Linear Factorization Theorem to find the polynomial function.
Find a fourth degree polynomial with real coefficients that has zeros of –3, 2,such that
Because is a zero, by the Complex Conjugate Theorem is also a zero. The polynomial must have factors ofandSince we are looking for a degree 4 polynomial, and now have four zeros, we have all four factors. Let’s begin by multiplying these factors.
We need to find a to ensureSubstituteand into
So the polynomial function is
or
We found that bothandwere zeros, but only one of these zeros needed to be given. Ifis a zero of a polynomial with real coefficients, thenmust also be a zero of the polynomial becauseis the complex conjugate of
If were given as a zero of a polynomial with real coefficients, would also need to be a zero?
Yes. When any complex number with an imaginary component is given as a zero of a polynomial with real coefficients, the conjugate must also be a zero of the polynomial.
Find a third degree polynomial with real coefficients that has zeros of 5 andsuch that
There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes’ Rule of Signs tells us of a relationship between the number of sign changes inand the number of positive real zeros. For example, the polynomial function below has one sign change.
This tells us that the function must have 1 positive real zero.
There is a similar relationship between the number of sign changes inand the number of negative real zeros.
In this case,has 3 sign changes. This tells us thatcould have 3 or 1 negative real zeros.
According to Descartes’ Rule of Signs, if we let be a polynomial function with real coefficients:
Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros for
Begin by determining the number of sign changes.
There are two sign changes, so there are either 2 or 0 positive real roots. Next, we examineto determine the number of negative real roots.
Again, there are two sign changes, so there are either 2 or 0 negative real roots.
There are four possibilities, as we can see in [link].
Positive Real Zeros | Negative Real Zeros | Complex Zeros | Total Zeros |
---|---|---|---|
2 | 2 | 0 | 4 |
2 | 0 | 2 | 4 |
0 | 2 | 2 | 4 |
0 | 0 | 4 | 4 |
We can confirm the numbers of positive and negative real roots by examining a graph of the function. See [link]. We can see from the graph that the function has 0 positive real roots and 2 negative real roots.
Use Descartes’ Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for Use a graph to verify the numbers of positive and negative real zeros for the function.
There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.
We have now introduced a variety of tools for solving polynomial equations. Let’s use these tools to solve the bakery problem from the beginning of the section.
A new bakery offers decorated sheet cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?
Begin by writing an equation for the volume of the cake. The volume of a rectangular solid is given byWe were given that the length must be four inches longer than the width, so we can express the length of the cake asWe were given that the height of the cake is one-third of the width, so we can express the height of the cake asLet’s write the volume of the cake in terms of width of the cake.
Substitute the given volume into this equation.
Descartes' rule of signs tells us there is one positive solution. The Rational Zero Theorem tells us that the possible rational zeros are and We can use synthetic division to test these possible zeros. Only positive numbers make sense as dimensions for a cake, so we need not test any negative values. Let’s begin by testing values that make the most sense as dimensions for a small sheet cake. Use synthetic division to check
Since 1 is not a solution, we will check
Since 3 is not a solution either, we will test
Synthetic division gives a remainder of 0, so 9 is a solution to the equation. We can use the relationships between the width and the other dimensions to determine the length and height of the sheet cake pan.
The sheet cake pan should have dimensions 13 inches by 9 inches by 3 inches.
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.
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.
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.
For the following exercises, use the Rational Zero Theorem to find all real zeros.
For the following exercises, find all complex solutions (real and non-real).
For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.
1 positive, 1 negative
3 or 1 positive, 0 negative
0 positive, 3 or 1 negative
2 or 0 positive, 2 or 0 negative
2 or 0 positive, 2 or 0 negative
For the following exercises, list all possible rational zeros for the functions.
For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.
For the following exercises, construct a polynomial function of least degree possible using the given information.
Real roots: –1, 1, 3 and
Real roots: –1 (with multiplicity 2 and 1) and
Real roots: –2, (with multiplicity 2) and
Real roots: , 0, and
Real roots: –4, –1, 1, 4 and
For the following exercises, find the dimensions of the box described.
The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.
8 by 4 by 6 inches
The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.
The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.
5.5 by 4.5 by 3.5 inches
The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.
The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.
8 by 5 by 3 inches
For the following exercises, find the dimensions of the right circular cylinder described.
The radius is 3 inches more than the height. The volume iscubic meters.
The height is one less than one half the radius. The volume iscubic meters.
Radius = 6 meters, Height = 2 meters
The radius and height differ by one meter. The radius is larger and the volume iscubic meters.
The radius and height differ by two meters. The height is greater and the volume is cubic meters.
Radius = 2.5 meters, Height = 4.5 meters
The radius ismeter greater than the height. The volume is cubic meters.