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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

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
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if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
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Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
Give me the reciprocal of even number
The reciprocal of an even number is a proper fraction
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
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archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc 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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