<< Chapter < Page Chapter >> Page >

A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?

3 meters by 4 meters by 7 meters

Got questions? Get instant answers now!

Key concepts

  • To find f ( k ) , determine the remainder of the polynomial f ( x ) when it is divided by x k . This is known as the Remainder Theorem. See [link] .
  • According to the Factor Theorem, k is a zero of f ( x ) if and only if ( x k ) is a factor of f ( x ) . See [link] .
  • According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See [link] and [link] .
  • When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
  • Synthetic division can be used to find the zeros of a polynomial function. See [link] .
  • According to the Fundamental Theorem, every polynomial function has at least one complex zero. See [link] .
  • Every polynomial function with degree greater than 0 has at least one complex zero.
  • Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form ( x c ) , where c is a complex number. See [link] .
  • The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
  • The number of negative real zeros of a polynomial function is either the number of sign changes of f ( x ) or less than the number of sign changes by an even integer. See [link] .
  • Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See [link] .

Section exercises

Verbal

Describe a use for the Remainder Theorem.

The theorem can be used to evaluate a polynomial.

Got questions? Get instant answers now!

Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

Got questions? Get instant answers now!

What is the difference between rational and real zeros?

Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

Got questions? Get instant answers now!

If Descartes’ Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?

Got questions? Get instant answers now!

If synthetic division reveals a zero, why should we try that value again as a possible solution?

Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

Got questions? Get instant answers now!

Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.

( x 4 9 x 2 + 14 ) ÷ ( x 2 )

Got questions? Get instant answers now!

( 3 x 3 2 x 2 + x 4 ) ÷ ( x + 3 )

106

Got questions? Get instant answers now!

( x 4 + 5 x 3 4 x 17 ) ÷ ( x + 1 )

Got questions? Get instant answers now!

( 3 x 2 + 6 x + 24 ) ÷ ( x 4 )

0

Got questions? Get instant answers now!

( 5 x 5 4 x 4 + 3 x 3 2 x 2 + x 1 ) ÷ ( x + 6 )

Got questions? Get instant answers now!

Questions & Answers

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
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
madras university algebra questions papers first year B. SC. maths
Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
y=x+1
gary
x=_1
Daulat
yas. x= -4
Deepak
x=-1
Deepak
2x=3-5 x=-2/2=-1
Rukmini
-1
Bobmorris
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