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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

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Key concepts

  • To find f ( k ) , determine the remainder of the polynomial f ( x ) when it is divided by x k . See [link] .
  • k is a zero of f ( x ) if and only if ( x k ) is a factor of f ( x ) . See [link] .
  • 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.

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

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What is the difference between rational and real zeros?

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

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

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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.

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Algebraic

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

( x 4 9 x 2 + 14 ) ÷ ( x 2 )

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( 3 x 3 2 x 2 + x 4 ) ÷ ( x + 3 )

106

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( x 4 + 5 x 3 4 x 17 ) ÷ ( x + 1 )

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( 3 x 2 + 6 x + 24 ) ÷ ( x 4 )

0

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

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( x 4 1 ) ÷ ( x 4 )

255

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( 3 x 3 + 4 x 2 8 x + 2 ) ÷ ( x 3 )

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( 4 x 3 + 5 x 2 2 x + 7 ) ÷ ( x + 2 )

1

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For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

Questions & Answers

how does the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
did you g8ve certificate
Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
hllo
John
hi
John
qt rrt
John
r u there
John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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