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1.4 Polynomials  (Page 2/15)

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Identifying the degree and leading coefficient of a polynomial

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

  1. 3 + 2 x 2 − 4 x 3
  2. 5 t 5 − 2 t 3 + 7 t
  3. 6 p − p 3 − 2
  1. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree,   −4 x 3 .   The leading coefficient is the coefficient of that term,   −4.
  2. The highest power of t is   5 , so the degree is   5.   The leading term is the term containing that degree,   5 t 5 .   The leading coefficient is the coefficient of that term,   5.
  3. The highest power of p is   3 , so the degree is   3.   The leading term is the term containing that degree,   − p 3 , The leading coefficient is the coefficient of that term,   −1.
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Identify the degree, leading term, and leading coefficient of the polynomial   4 x 2 − x 6 + 2 x − 6.

The degree is 6, the leading term is   − x 6 , and the leading coefficient is   −1.

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Adding and subtracting polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example,   5 x 2   and   −2 x 2   are like terms, and can be added to get   3 x 2 , but   3 x   and   3 x 2   are not like terms, and therefore cannot be added.

Given multiple polynomials, add or subtract them to simplify the expressions.

  1. Combine like terms.
  2. Simplify and write in standard form.

Adding polynomials

Find the sum.

( 12 x 2 + 9 x − 21 ) + ( 4 x 3 + 8 x 2 − 5 x + 20 )

4 x 3 + ( 12 x 2 + 8 x 2 ) + ( 9 x − 5 x ) + ( −21 + 20 )      Combine like terms . 4 x 3 + 20 x 2 + 4 x − 1    Simplify .

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Find the sum.

( 2 x 3 + 5 x 2 − x + 1 ) + ( 2 x 2 − 3 x − 4 )

2 x 3 + 7 x 2 −4 x −3

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Subtracting polynomials

Find the difference.

( 7 x 4 − x 2 + 6 x + 1 ) − ( 5 x 3 − 2 x 2 + 3 x + 2 )

7 x 4 − 5 x 3 + ( − x 2 + 2 x 2 ) + ( 6 x − 3 x ) + ( 1 − 2 )    Combine like terms . 7 x 4 − 5 x 3 + x 2 + 3 x − 1 Simplify .

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Find the difference.

( −7 x 3 − 7 x 2 + 6 x − 2 ) − ( 4 x 3 − 6 x 2 − x + 7 )

−11 x 3 − x 2 + 7 x −9

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Multiplying polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

Multiplying polynomials using the distributive property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the   2   in   2 ( x + 7 )   to obtain the equivalent expression   2 x + 14.   When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

Given the multiplication of two polynomials, use the distributive property to simplify the expression.

  1. Multiply each term of the first polynomial by each term of the second.
  2. Combine like terms.
  3. Simplify.

Multiplying polynomials using the distributive property

Find the product.

( 2 x + 1 ) ( 3 x 2 − x + 4 )

2 x ( 3 x 2 − x + 4 ) + 1 ( 3 x 2 − x + 4 )      Use the distributive property . ( 6 x 3 − 2 x 2 + 8 x ) + ( 3 x 2 − x + 4 )    Multiply . 6 x 3 + ( −2 x 2 + 3 x 2 ) + ( 8 x − x ) + 4    Combine like terms . 6 x 3 + x 2 + 7 x + 4      Simplify .

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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