<< Chapter < Page Chapter >> Page >

Given a polynomial function f ( x ) , use the Rational Zero Theorem to find rational zeros.

  1. Determine all factors of the constant term and all factors of the leading coefficient.
  2. Determine all possible values of p q , where p is a factor of the constant term and q is a factor of the leading coefficient. Be sure to include both positive and negative candidates.
  3. Determine which possible zeros are actual zeros by evaluating each case of f ( p q ) .

Listing all possible rational zeros

List all possible rational zeros of f ( x ) = 2 x 4 5 x 3 + x 2 4.

The only possible rational zeros of f ( 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 are p = ±1 , ±2 , ±4.

The leading coefficient is 2; the factors of 2 are q = ±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.

p q = ± 1 1 , ± 1 2      p q = ± 2 1 , ± 2 2      p q = ± 4 1 , ± 4 2

Note that 2 2 = 1 and 4 2 = 2 , which have already been listed. So we can shorten our list.

p q = Factors of the last Factors of the first = ±1 , ±2 , ±4 , ± 1 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using the rational zero theorem to find rational zeros

Use the Rational Zero Theorem to find the rational zeros of f ( x ) = 2 x 3 + x 2 4 x + 1.

The Rational Zero Theorem tells us that if p q is a zero of f ( x ) , then p is a factor of 1 and q is a factor of 2.

p q = factor of constant term factor of leading coefficient     = factor of 1 factor of 2

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

   f ( 1 ) = 2 ( 1 ) 3 + ( 1 ) 2 4 ( 1 ) + 1 = 4        f ( 1 ) = 2 ( 1 ) 3 + ( 1 ) 2 4 ( 1 ) + 1 = 0     f ( 1 2 ) = 2 ( 1 2 ) 3 + ( 1 2 ) 2 4 ( 1 2 ) + 1 = 3        f ( 1 2 ) = 2 ( 1 2 ) 3 + ( 1 2 ) 2 4 ( 1 2 ) + 1 = 1 2

Of those, −1, 1 2 ,  and  1 2 are not zeros of f ( x ) . 1 is the only rational zero of f ( x ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the Rational Zero Theorem to find the rational zeros of f ( x ) = x 3 5 x 2 + 2 x + 1.

There are no rational zeros.

Got questions? Get instant answers now!

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 function f , use synthetic division to find its zeros.

  1. Use the Rational Zero Theorem to list all possible rational zeros of the function.
  2. 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.
  3. Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.
  4. 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 of f ( x ) = 4 x 3 3 x 1.

The Rational Zero Theorem tells us that if p q is a zero of f ( x ) , then p is a factor of –1 and q is a factor of 4.

p q = factor of constant term factor of leading coefficient     = factor of –1 factor of 4

The factors of 1 are ±1 and the factors of 4 are ±1 , ±2 , and ±4. The possible values for p q are ±1 , ± 1 2 , and ± 1 4 . 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.

1 4 0 3 1 4 4 1    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 ) ( 4 x 2 + 4 x + 1 ) .

The quadratic is a perfect square. f ( x ) can be written as

( x 1 ) ( 2 x + 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.

2 x + 1 = 0           x = 1 2

The zeros of the function are 1 and 1 2 with multiplicity 2.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask