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In this section, you will:
  • Evaluate a polynomial using the Remainder Theorem.
  • Use the Factor Theorem to solve a polynomial equation.
  • Use the Rational Zero Theorem to find rational zeros.
  • Find zeros of a polynomial function.
  • Use the Linear Factorization Theorem to find polynomials with given zeros.
  • Use Descartes’ Rule of Signs.
  • Solve real-world applications of 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?

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

f ( k ) = ( k k ) q ( k ) + r = 0 q ( k ) + r = r

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

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 ) .

Using the remainder theorem to evaluate a polynomial

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

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

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

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

f ( 3 ) = 412

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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.

f ( x ) = ( x k ) q ( x ) + r

If k is a zero, then the remainder r 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 ) .

Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 6

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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