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  5. Factoring trinomials with leading

1.5 Factoring trinomials with leading coefficient other than 1

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses.The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: be able to factor trinomials with leading coefficient other than 1.

Overview

  • The Method of Factorization

The method of factorization

In the last section we saw that we could easily factor trinomials of the form x 2 b x c by finding the factors of the constant c that add to the coefficient of the linear term b , as shown in the following example:

Factor x 2 - 4 x - 21 .
The third term of the trinomial is − 21 . We seek two numbers whose

(a) product is − 21 and
(b) sum is − 4 .

The required numbers are − 7 and + 3 .

x 2 - 4 x - 21 ( x - 7 ) ( x 3 )

The problem of factoring the polynomial a x 2 b x c , a≠1 , becomes more involved. We will study two methods of factoring such polynomials. Each method produces the same result, and you should select the method you are most comfortable with. The first method is called the trial and error method and requires some educated guesses. We will examine two examples (Sample Sets A and B). Then, we will study a second method of factoring. The second method is called the collect and discard method , and it requires less guessing than the trial and error method. Sample Set C illustrates the use of the collect and discard method.

The trial and error method of factoring a x 2 + b x + c

Trial and error method

Consider the product

Steps showing the product of two binomials 'four x plus three,' and 'five x plus two.' See the longdesc for a full description.

Examining the trinomial 20 x 2 23 x 6 , we can immediately see some factors of the first and last terms.

20 x 2 6
20 x ,     x 6 ,     1
10 x ,     2 x 3 ,     2
    5 x ,     4 x


Our goal is to choose the proper combination of factors of the first and last terms that yield the middle term 23 x .
Notice that the middle term comes from the sum of the outer and inner products in the multiplication of the two binomials.
The product of two binomials four x plus three, and five x plus two. The outer product of binomials is eight x, and the inner product is fifteen x.
This fact provides us a way to find the proper combination.

Look for the combination that when multiplied and then added yields the middle term.

The proper combination we're looking for is

The product of the first and the last term is twenty x squared. One of the combinations of the factors of the first and last term yields two new factors of the product such that their sum is the middle term: twenty three x.

Sample set a

Factor 6 x 2 + x − 12 .

Factor the first and last terms.

The factors of the first term 'six x squared' and the last term 'negative twelve' are shown. The product of the first and the last term is negative seventy-two x squared. One of the combinations of the factors of the first and the last term yields two new factors of the product such that their sum is the middle term: x.

Thus, 3 x and 3 are to be multiplied, 2 x and − 4 are to be multiplied.

6 x 2 + x − 12 = ( ) ( ) Put   the   factors   of   the   leading   term   in   immediately .   = ( 3 x ) ( 2 x ) Since   3 x   and   3   are   to   be   multiplied,   they   must   be   located   in   different   binomials . = ( 3 x ) ( 2 x + 3 ) Place   the   − 4   in   the   remaining   set   of   parentheses . = ( 3 x − 4 ) ( 2 x + 3 ) 6 x 2 + x − 12 = ( 3 x − 4 ) ( 2 x + 3 )

C h e c k :       ( 3 x − 4 ) ( 2 x + 3 ) = 6 x 2 + 9 x − 8 x − 12 = 6 x 2 + x − 12

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, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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