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Mathematics LibreTexts

3.6: Zeros of Polynomial Functions

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.

Evaluating a Polynomial Using the Remainder Theorem

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 by \(x–k\),

the remainder may be found quickly by evaluating the polynomial function at \(k\), that is, \(f(k)\). Let’s walk through the proof of the theorem.

Recall that the Division Algorithm states that, given a polynomial dividend \(f(x)\) and a non-zero polynomial divisor \(d(x)\) where the degree of \(d(x)\) is less than or equal to the degree of \(f(x)\),there exist unique polynomials \(q(x)\) and \(r(x)\) such that

\[f(x)=d(x)q(x)+r(x)\]

If the divisor, \(d(x)\), is \(x−k\), this takes the form

\[f(x)=(x−k)q(x)+r\]

Since the divisor \(x−k\)

is linear, the remainder will be a constant, \(r\). And, if we evaluate this for \(x=k\), we have

\[\begin{align} f(k)&=(k−k)q(k)+r \\ &=0{\cdot}q(k)+r \\ &=r \end{align}\]

In other words, \(f(k)\) is the remainder obtained by dividing \(f(x)\)by \(x−k\).

Note The Remainder Theorem:

If a polynomial \(f(x)\) is divided by \(x−k\),then the remainder is the value \(f(k)\).

Given a polynomial function \(f\), evaluate \(f(x)\) at \(x=k\) using the Remainder Theorem.

  1. Use synthetic division to divide the polynomial by \(x−k\).
  2. The remainder is the value \(f(k)\).

Example 3.6.1: Using the Remainder Theorem to Evaluate a Polynomial

Use the Remainder Theorem to evaluate \(f(x)=6x^4−x^3−15x^2+2x−7\) at \(x=2\).

Solution

To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by \(x−2\).

26−1−152−712  221432   611    71625

The remainder is 25. Therefore, \(f(2)=25\).

Analysis

We can check our answer by evaluating \(f(2)\).

\[\begin{align} f(x)&=6x^4−x^3−15x^2+2x−7 \\ f(2)&=6(2)4−(2)3−15(2)2+2(2)−7 \\ &=25 \end{align}\]

3.6.1: Use the Remainder Theorem to evaluate \(f(x)=2x^5−3x^4−9x^3+8x^2+2\) at \(x=−3\).

Solution

\(f(−3)=−412\)

Using the Factor Theorem to Solve a Polynomial Equation

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 tells us

\[f(x)=(x−k)q(x)+r\].

If \(k\) is a zero, then the remainderr is \(f(k)=0\) and \(f(x)=(x−k)q(x)+0\) or \(f(x)=(x−k)q(x)\).

Notice, written in this form, \(x−k\) is a factor of \(f(x)\).We can conclude if \(k\) is a zero of \(f(x)\),then \(x−k\) is a factor of \(f(x)\).

Similarly, if \(x−k\) is a factor of \(f(x)\), then the remainder of the Division Algorithm \(f(x)=(x−k)q(x)+r\) is 0. This tells us that \(k\)

is a zero.

This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree \(n\) in the complex number system will haven zeros. We can use the Factor Theorem to completely factor a polynomial into the product of \(n\) factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

Note: The Factor Theorem

According to the Factor Theorem, \(k\) is a zero of \(f(x)\) if and only if \((x−k)\) is a factor of \(f(x)\).

Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.

  • Use synthetic division to divide the polynomial by \((x−k)\).
  • Confirm that the remainder is 0.
  • Write the polynomial as the product of(x−k)
  • and the quadratic quotient.
  • If possible, factor the quadratic.
  • Write the polynomial as the product of factors.

Using the Factor Theorem to Solve a Polynomial Equation

Show that(x+2)

is a factor ofx3−6x2−x+30.

Find the remaining factors. Use the factors to determine the zeros of the polynomial.

We can use synthetic division to show that(x+2)

is a factor of the polynomial.


−21−6−130−216−30      1−815    0

The remainder is zero, so(x+2)

is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:

(x+2)(x2−8x+15)

We can factor the quadratic factor to write the polynomial as

(x+2)(x−3)(x−5)

By the Factor Theorem, the zeros ofx3−6x2−x+30

are –2, 3, and 5.

Use the Factor Theorem to find the zeros off(x)=x3+4x2−4x−16

given that(x−2)

is a factor of the polynomial.

The zeros are 2, –2, and –4.

Using the Rational Zero Theorem to Find Rational Zeros

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,x=25

andx=34.

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

The Rational Zero Theorem states that, if the polynomialf(x)=anxn+an−1xn−1+...+a1x+a0

has integer coefficients, then every rational zero off(x) has the formpqwherepis a factor of the constant terma0andqis a factor of the leading coefficientan.

When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.

Given a polynomial functionf(x),

use the Rational Zero Theorem to find rational zeros.

Determine all factors of the constant term and all factors of the leading coefficient.

Determine all possible values ofpq,

wherepis a factor of the constant term andq

is a factor of the leading coefficient. Be sure to include both positive and negative candidates.

Determine which possible zeros are actual zeros by evaluating each case off(pq).


Listing All Possible Rational Zeros

List all possible rational zeros off(x)=2x4−5x3+x2−4.

The only possible rational zeros off(x)

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 arep=±1,±2,±4.

The leading coefficient is 2; the factors of 2 areq=±1,±2.

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.

pq=±11,±12    pq=±21,±22    pq=±41,±42

Note that22=1

and42=2,

which have already been listed. So we can shorten our list.

pq=Factors of the lastFactors of the first=±1,±2,±4,±12

Using the Rational Zero Theorem to Find Rational Zeros

Use the Rational Zero Theorem to find the rational zeros off(x)=2x3+x2−4x+1.

The Rational Zero Theorem tells us that ifpq

is a zero off(x), thenpis a factor of 1 andq

is a factor of 2.

pq=factor of constant termfactor of leading coefficient   =factor of 1factor of 2

The factors of 1 are±1

and the factors of 2 are±1and±2.The possible values forpqare±1and±12.These are the possible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting these values forxinf(x).

  \(f(−1)=2(−1)3+(−1)2−4(−1)+1=4\)      \(f(1)=2(1)3+(1)2−4(1)+1=0\)   \(f(−12)=2(−12)3+(−12)2−4(−12)+1=3\)      \(f(12)=2(12)3+(12)2−4(12)+1=−12\)

Of those, −1,−12, and 12

are not zeros off(x).1 is the only rational zero off(x).

Use the Rational Zero Theorem to find the rational zeros off(x)=x3−5x2+2x+1.

There are no rational zeros.

Finding the Zeros of Polynomial Functions

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 functionf,

use synthetic division to find its zeros.

Use the Rational Zero Theorem to list all possible rational zeros of the function.

Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.

Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.

Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.

Finding the Zeros of a Polynomial Function with Repeated Real Zeros

Find the zeros off(x)=4x3−3x−1.

The Rational Zero Theorem tells us that ifpq

is a zero off(x),thenp is a factor of –1 andq

is a factor of 4.

pq=factor of constant termfactor of leading coefficient   =factor of –1factor of 4

The factors of–1

are±1 and the factors of4 are±1,±2,and±4.The possible values forpqare±1,±12,and±14.

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.

140−3−1441   4 4  1   0

Dividing by(x−1)

gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as

(x−1)(4x2+4x+1).

The quadratic is a perfect square.f(x)

can be written as

(x−1)(2x+1)2.

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.

2x+1=0         \(x=−12\)

The zeros of the function are 1 and−12

with multiplicity 2.

Analysis

Look at the graph of the functionf

in Figure. Notice, atx=−0.5,the graph bounces off the x-axis, indicating the even multiplicity (2,4,6…) for the zero−0.5. Atx=1,the graph crosses the x-axis, indicating the odd multiplicity (1,3,5…) for the zerox=1.


Using the Fundamental Theorem of Algebra

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.

Supposef

is a polynomial function of degree four, andf(x)=0.The Fundamental Theorem of Algebra states that there is at least one complex solution, call itc1.By the Factor Theorem, we can writef(x)as a product ofx−c1and a polynomial quotient. Sincex−c1is 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 itc2.So we can write the polynomial quotient as a product ofx−c2and 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 off(x).

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, iff(x)

is a polynomial of degreen>0,anda is a non-zero real number, thenf(x)has exactlyn

linear factors

\(f(x)=a(x−c1)(x−c2)\)...(x−cn)

wherec1,c2,...,cn

are complex numbers. Therefore,f(x)hasn

roots if we allow for multiplicities.

Does every polynomial have at least one imaginary zero?

No. A complex number is not necessarily imaginary. Real numbers are also complex numbers.

Finding the Zeros of a Polynomial Function with Complex Zeros

Find the zeros off(x)=3x3+9x2+x+3.

The Rational Zero Theorem tells us that ifpq

is a zero off(x),thenpis a factor of 3 andq

is a factor of 3.

pq=factor of constant termfactor of leading coefficient   =factor of 3factor of 3

The factors of 3 are±1

and ±3.The possible values forpq,and therefore the possible rational zeros for the function, are ±3,±1, and ±13.

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.

−33913−90−3     3   0 1  0

Dividing by(x+3)

gives a remainder of 0, so –3 is a zero of the function. The polynomial can be written as

(x+3)(3x2+1)

We can then set the quadratic equal to 0 and solve to find the other zeros of the function.

3x2+1=0       x2=−13        \(x=±√−13=±i√33\)

The zeros off(x)

are –3 and±i√33.

Analysis

Look at the graph of the functionf

in Figure. Notice that, atx=−3,the graph crosses the x-axis, indicating an odd multiplicity (1) for the zerox=–3.Also 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 ofx=−3is 1 and there are two complex solutions, which is what we found, or the multiplicity atx=−3

is three. Either way, our result is correct.


Find the zeros off(x)=2x3+5x2−11x+4.

The zeros are–4, 12, and 1.

Using the Linear Factorization Theorem to Find Polynomials with Given Zeros

A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degreen

will havenzeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function inton 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 form(x−c),wherec

is a complex number.

Letf

be a polynomial function with real coefficients, and supposea+bi, b≠0, is a zero off(x). Then, by the Factor Theorem,x−(a+bi) is a factor off(x). Forf to have real coefficients,x−(a−bi) must also be a factor off(x). This is true because any factor other thanx−(a−bi), when multiplied byx−(a+bi), 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 functionf with real coefficients has a complex zeroa+bi, then the complex conjugatea−bi must also be a zero off(x).

This is called the Complex Conjugate Theorem.

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(x−c),

wherec

is a complex number.

If the polynomial functionf

has real coefficients and a complex zero in the forma+bi, then the complex conjugate of the zero,a−bi,

is also a zero.

Given the zeros of a polynomial functionf

and a point (c, \(f(c))\) on the graph off,

use the Linear Factorization Theorem to find the polynomial function.

Use the zeros to construct the linear factors of the polynomial.

Multiply the linear factors to expand the polynomial.

Substitute(c,f(c))

into the function to determine the leading coefficient.

Simplify.

Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros

Find a fourth degree polynomial with real coefficients that has zeros of –3, 2,i,

such thatf(−2)=100.

Becausex=i

is a zero, by the Complex Conjugate Theoremx=–i is also a zero. The polynomial must have factors of(x+3),(x−2),(x−i),and(x+i).

Since 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.

\(f(x)=a(x+3)(x−2)(x−i)(x+i)f(x)=a(x2+x−6)(x2+1)f(x)=a(x4+x3−5x2+x−6)\)

We need to find a to ensuref(–2)=100.

Substitutex=–2 andf(2)=100 intof(x).

100=a((−2)4+(−2)3−5(−2)2+(−2)−6)100=a(−20)−5=a

So the polynomial function is

\(f(x)=−5(x4+x3−5x2+x−6)\)

or

\(f(x)=−5x4−5x3+25x2−5x+30\)

Analysis

We found that bothi

and−iwere zeros, but only one of these zeros needed to be given. Ifiis a zero of a polynomial with real coefficients, then−imust also be a zero of the polynomial because−iis the complex conjugate ofi.

If2+3i

were given as a zero of a polynomial with real coefficients, would 2−3i

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 and−2i

such thatf(1)=10.

\(f(x)=−12x3+52x2−2x+10\)

Using Descartes’ Rule of Signs

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 inf(x)

and 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 inf(−x)

and the number of negative real zeros.


In this case,f(−x)

has 3 sign changes. This tells us thatf(x)

could have 3 or 1 negative real zeros.

Descartes’ Rule of Signs

According to Descartes’ Rule of Signs, if we let \(f(x)=anxn+an−1xn−1+\)...+a1x+a0

be a polynomial function with real coefficients:

The number of positive real zeros is either equal to the number of sign changes off(x)

or is less than the number of sign changes by an even integer.

The number of negative real zeros is either equal to the number of sign changes off(−x)

or is less than the number of sign changes by an even integer.

Using Descartes’ Rule of Signs

Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros forf(x)=−x4−3x3+6x2−4x−12.

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 examinef(−x)

to determine the number of negative real roots.

\(f(−x)=−(−x)4−3(−x)3+6(−x)2−4(−x)−12f(−x)=−x4+3x3+6x2+4x−12\)


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 Table.


Positive Real ZerosNegative Real ZerosComplex ZerosTotal Zeros

2204

2024

0224

0044

Analysis

We can confirm the numbers of positive and negative real roots by examining a graph of the function. See Figure. 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 forf(x)=2x4−10x3+11x2−15x+12.

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.

Solving Real-World Applications

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.

Solving Polynomial Equations

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 byV=lwh.

We were given that the length must be four inches longer than the width, so we can express the length of the cake asl=w+4.We were given that the height of the cake is one-third of the width, so we can express the height of the cake ash=13w.

Let’s write the volume of the cake in terms of width of the cake.

\(V=(w+4)(w)(13w)V=13w3+43w2\)

Substitute the given volume into this equation.

  351=13w3+43w2Substitute 351 for V.1053=w3+4w2Multiply both sides by 3.      0=w3+4w2−1053 Subtract 1053 from both sides.

Descartes' rule of signs tells us there is one positive solution. The Rational Zero Theorem tells us that the possible rational zeros are ±3,±9,±13,±27,±39,±81,±117,±351,

and±1053. 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 checkx=1.

1140−1053155  1 55−1048

Since 1 is not a solution, we will checkx=3.


Since 3 is not a solution either, we will testx=9.


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.

\(l=w+4=9+4=13\) and \(h=13w=13(9)=3\)

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.

Real Zeros, Factors, and Graphs of Polynomial Functions

Complex Factorization Theorem

Find the Zeros of a Polynomial Function

Find the Zeros of a Polynomial Function 2

Find the Zeros of a Polynomial Function 3

Key Concepts

To findf(k),

determine the remainder of the polynomialf(x)when it is divided byx−k.

See Example.

\(k\)

is a zero off(x) if and only if(x−k) is a factor off(x).

See Example.

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 Example and Example.

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 Example.

According to the Fundamental Theorem, every polynomial function has at least one complex zero. See Example.

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(x−c),

wherec

is a complex number. See Example.

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 off(−x)

or less than the number of sign changes by an even integer. See Example.

Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See Example.

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.

(x4−9x2+14)÷(x−2)

(3x3−2x2+x−4)÷(x+3)

−106

(x4+5x3−4x−17)÷(x+1)

(−3x2+6x+24)÷(x−4)

0

(5x5−4x4+3x3−2x2+x−1)÷(x+6)

(x4−1)÷(x−4)

255

(3x3+4x2−8x+2)÷(x−3)

(4x3+5x2−2x+7)÷(x+2)

−1

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

\(f(x)=2x3−9x2+13x−6;\) x−1

\(f(x)=2x3+x2−5x+2;\) x+2

−2, 1, 12

\(f(x)=3x3+x2−20x+12;\) x+3

\(f(x)=2x3+3x2+x+6;x+2\)

−2

\(f(x)=−5x3+16x2−9;x−3\)

x3+3x2+4x+12;x+3

−3

4x3−7x+3;x−1

2x3+5x2−12x−30,2x+5

−52, √6, −√6

For the following exercises, use the Rational Zero Theorem to find all real zeros.

x3−3x2−10x+24=0

2x3+7x2−10x−24=0

2, −4, −32

x3+2x2−9x−18=0

x3+5x2−16x−80=0

4, −4, −5

x3−3x2−25x+75=0

2x3−3x2−32x−15=0

5, −3, −12

2x3+x2−7x−6=0

2x3−3x2−x+1=0

12, 1+√52, 1−√52

3x3−x2−11x−6=0

2x3−5x2+9x−9=0

32

2x3−3x2+4x+3=0

x4−2x3−7x2+8x+12=0

2, 3, −1, −2

x4+2x3−9x2−2x+8=0

4x4+4x3−25x2−x+6=0

12, −12, 2, −3

2x4−3x3−15x2+32x−12=0

x4+2x3−4x2−10x−5=0

−1, −1, √5, −√5

4x3−3x+1=0

8x4+26x3+39x2+26x+6

−34, −12

For the following exercises, find all complex solutions (real and non-real).

x3+x2+x+1=0

x3−8x2+25x−26=0

2, 3+2i, 3−2i

x3+13x2+57x+85=0

3x3−4x2+11x+10=0

−23, 1+2i, 1−2i

x4+2x3+22x2+50x−75=0

2x3−3x2+32x+17=0

−12, 1+4i, 1−4i

Graphical

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

\(f(x)=x3−1\)

\(f(x)=x4−x2−1\)

1 positive, 1 negative


\(f(x)=x3−2x2−5x+6\)

\(f(x)=x3−2x2+x−1\)

3 or 1 positive, 0 negative


\(f(x)=x4+2x3−12x2+14x−5\)

\(f(x)=2x3+37x2+200x+300\)

0 positive, 3 or 1 negative


\(f(x)=x3−2x2−16x+32\)

\(f(x)=2x4−5x3−5x2+5x+3\)

2 or 0 positive, 2 or 0 negative


\(f(x)=2x4−5x3−14x2+20x+8\)

\(f(x)=10x4−21x2+11\)

2 or 0 positive, 2 or 0 negative


Numeric

For the following exercises, list all possible rational zeros for the functions.

\(f(x)=x4+3x3−4x+4\)

\(f(x)=2x3+3x2−8x+5\)

±5, ±1, ±52

\(f(x)=3x3+5x2−5x+4\)

\(f(x)=6x4−10x2+13x+1\)

±1, ±12, ±13, ±16

\(f(x)=4x5−10x4+8x3+x2−8\)

Technology

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.

\(f(x)=6x3−7x2+1\)

1, 12, −13

\(f(x)=4x3−4x2−13x−5\)

\(f(x)=8x3−6x2−23x+6\)

2, 14, −32

\(f(x)=12x4+55x3+12x2−117x+54\)

\(f(x)=16x4−24x3+x2−15x+25\)

54

Extensions

For the following exercises, construct a polynomial function of least degree possible using the given information.

Real roots: –1, 1, 3 and(2,f(2))=(2,4)

Real roots: –1 (with multiplicity 2 and 1) and(2,f(2))=(2,4)

\(f(x)=49(x3+x2−x−1)\)

Real roots: –2,12

(with multiplicity 2) and(−3,f(−3))=(−3,5)

Real roots:−12

, 0,12 and(−2,f(−2))=(−2,6)

\(f(x)=−15(4x3−x)\)

Real roots: –4, –1, 1, 4 and(−2,f(−2))=(−2,10)

Real-World Applications

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 is16π

cubic meters.

The height is one less than one half the radius. The volume is72π

cubic meters.

Radius = 6 meters, Height = 2 meters

The radius and height differ by one meter. The radius is larger and the volume is48π

cubic meters.

The radius and height differ by two meters. The height is greater and the volume is28.125π

cubic meters.

Radius = 2.5 meters, Height = 4.5 meters

80. The radius is13

meter greater than the height. The volume is989π

cubic meters.

Glossary

Descartes’ Rule of Signs

a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes off(x)

andf(−x)


Factor Theorem

\(k\)

is a zero of polynomial functionf(x)if and only if(x−k) is a factor off(x)


Fundamental Theorem of Algebra

a polynomial function with degree greater than 0 has at least one complex zero

Linear Factorization Theorem

allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form (x−c),

wherec

is a complex number

Rational Zero Theorem

the possible rational zeros of a polynomial function have the formpq

wherepis a factor of the constant term andq

is a factor of the leading coefficient.

Remainder Theorem

if a polynomialf(x)

is divided byx−k,then the remainder is equal to the valuef(k)