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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

answer and questions in exercise 11.2 sums
Yp Reply
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Jallah Reply
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
C'est comment
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it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
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
 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
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
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
Aasik Reply
Wrong question
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
2x + 7 =19
2x +7=19. 2x=19 - 7 2x=12 x=6
because x is 6
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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