<< Chapter < Page Chapter >> Page >
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.

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

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

Got questions? Get instant answers now!

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

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
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask