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1.5 Factoring polynomials  (Page 3/6)

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Factor a.   2 x 2 + 9 x + 9   b.   6 x 2 + x − 1

a.   ( 2 x + 3 ) ( x + 3 )   b.   ( 3 x −1 ) ( 2 x + 1 )

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Factoring a perfect square trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

a 2 + 2 a b + b 2 = ( a + b ) 2 and a 2 − 2 a b + b 2 = ( a − b ) 2

We can use this equation to factor any perfect square trinomial.

Perfect square trinomials

A perfect square trinomial can be written as the square of a binomial:

a 2 + 2 a b + b 2 = ( a + b ) 2

Given a perfect square trinomial, factor it into the square of a binomial.

  1. Confirm that the first and last term are perfect squares.
  2. Confirm that the middle term is twice the product of   a b .
  3. Write the factored form as   ( a + b ) 2 .

Factoring a perfect square trinomial

Factor   25 x 2 + 20 x + 4.

Notice that   25 x 2   and   4   are perfect squares because   25 x 2 = ( 5 x ) 2   and   4 = 2 2 .   Then check to see if the middle term is twice the product of   5 x   and   2.   The middle term is, indeed, twice the product:   2 ( 5 x ) ( 2 ) = 20 x .   Therefore, the trinomial is a perfect square trinomial and can be written as   ( 5 x + 2 ) 2 .

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Factor   49 x 2 − 14 x + 1.

( 7 x −1 ) 2

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Factoring a difference of squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

a 2 − b 2 = ( a + b ) ( a − b )

We can use this equation to factor any differences of squares.

Differences of squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

a 2 − b 2 = ( a + b ) ( a − b )

Given a difference of squares, factor it into binomials.

  1. Confirm that the first and last term are perfect squares.
  2. Write the factored form as   ( a + b ) ( a − b ) .

Factoring a difference of squares

Factor   9 x 2 − 25.

Notice that   9 x 2   and   25   are perfect squares because   9 x 2 = ( 3 x ) 2   and   25 = 5 2 .   The polynomial represents a difference of squares and can be rewritten as   ( 3 x + 5 ) ( 3 x − 5 ) .

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Factor   81 y 2 − 100.

( 9 y + 10 ) ( 9 y − 10 )

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Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Factoring the sum and difference of cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 )

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 )

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.

x 3 − 2 3 = ( x − 2 ) ( x 2 + 2 x + 4 )

The sign of the first 2 is the same as the sign between   x 3 − 2 3 .   The sign of the   2 x   term is opposite the sign between   x 3 − 2 3 .   And the sign of the last term, 4, is always positive .

Sum and difference of cubes

We can factor the sum of two cubes as

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 )

We can factor the difference of two cubes as

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 )

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
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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, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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