Grade 8: Factoring: General Trinomials
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Grade 8
MATH
MODULE 5:
FACTORING: GENERAL TRINOMIALS
MDM-Sagay College, Inc.
Office: Feliza Bldg., Marañon St. Pob 2, Sagay City
Campus: National Highway, Poblacion 2, Sagay City, Negros Occidental
Tel.# 488-0531/ email: mdm_sagay2000@gmail.com
Lesson 1: FACTORING: GENERAL TRINOMIALS
What`s In
A trinomial which is not a perfect square trinomial may be a quadratic trinomial in
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the form of ax +bx+c , where a, b, and c are constants and a≠0.
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There are two types of quadratic trinomial in the form ax +bx +c .
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1. ax +bx +c , a=1
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2. ax +bx +c , a≠1
Before we proceed to our examples let us first recall the rules of integers, for you to easily find the
factors of our trinomials.
Rules for Integers
ADDITION SUBTRACTION
(+) and (+) = (+)
(-) and (-) = (-) Change the sign of the subtrahend then
proceed to addition rules.
subtract, then copy Ex: 5 - (-4)
(+) and (-) = the sign of the Subtrahend is -4 , change it to +4
(-) and (+) = bigger number
MULTIPLICATION and DIVISION
(+) and (+) = (+) (+) and (-) = (-)
(-) and (-) = (+) (-) and (+) = (-)
What Is it
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Factoring trinomials ax +bx +c , where a= 1
We will study first how trinomials whose leading coefficient is 1 are being factored.
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1. Factor d +5 d +6
Solution: a. List all the possible factors of 6
Factors of 6
6 1
3 2
-6 -1
-3 -2
b. From your list of possible factors, find a pair whose sum is 5 (which is the middle
term).
6+ 1 =7 3 and 2 is the pair that we wanted to find whose sum is 5. Therefore,
3 +2 =5 the factors of d2+ 5d + 6 = (d + 3)(d + 2)
-6 + (-1) = -7
-3 + (-2) = -5
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2. Factor k + 3 k−28
a. List all the possible factors of -28 (Note: the sign of the number is negative (-), then
use a (+) and (-) pair of numbers.)
Factors of -28
7 -4
-7 4
14 -2
-14 2
28 -1
-28 1
b. Find a pair whose sum is 3( the middle term of the trinomial).
(check the rules of integers table)
7 + (-4) =3 7 and -4 is the pair that we wanted to
-7 + 4 =(-3)
find whose sum is 3. Thus, the factor of
14 + (-2) = 12
k2 + 3k – 28 = (k + 7)(k – 4)
-14 + 2 =-12
28 + (-1) = 27 (k +7 )(k −4 ). .
-28 + 1 = -27
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Factoring trinomials ax +bx +c , where a≠ 1
There are many ways of factoring these types of polynomials of this form, one of which is by
inspection. Trial and error method is utilized in factoring this type of trinomials.
1. Factor 2 q3 −6 q 2−36 q
Noticeable that there is a common monomial factor. Begin by factoring out 2q first.
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Rewriting it, you have 2 q(q −3 q−18 ).
Now we can follow the steps we did earlier, when a=1.
a. Listing all the factors of the 3rd term (-18) and finding a pair whose sum is equal to
the 2nd term which is (-3).
Factors of -18
18 -1
-18 1
9 -2
-9 2
6 -3
-6 3
b. Since -6 and 3 are the factors of -18 whose sum is -3, then the binomial factors of
(q2 −3 q−18 ) are (q−6 )(q+3 ).
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c. Therefore, the factors of 2 q −6q −36q are 2 q(q−6 )(q+3 ).
d.
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2. Factor 2 x + 3 x−5
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The factors of 2 x are 2x and x ; and of the third term -5 are (5)and (−1) ,
(−5)and (1) . Pair all possible binomial factors.
a. (2 x +5 )(x −1) c. (2 x +1)( x−5 )
b (2 x−5 )( x+1) d. (2 x−1 )( x+5 )
To know which from the four pairs above are the factors of the given trinomial, we must get the
product of the pair of binomials that is equal to the given expression.
Let’s Check: Use the FOIL Method you had learned from your grade 7 Mathematics.
FOIL (First, Outside, Inside, Last)
Arrange the products in the form of
ax2 + bx + c.
(2 x +5 )(x −1)
2x2 – 2x + 5x – 5
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First Terms (2 x )( x )= 2 x
−2 x (combine like terms, -2x + 5x)
Outside Terms (2x )(−1)=
Inside Terms (5)( x )= 5 x Thus, the product of (2x + 5)(x – 1) is, 2x2 +
Last Terms (5)(−1)= −5 3x – 5, then such pair of binomials are the
factors of the given expression.
Remember:
To factor trinomials with 1 as the numerical coefficient of the leading term:
a. Factor the leading term of the trinomial and write these factors as the leading terms of the factors;
b. List down all the factors of the last term;
c. Identify which factor pair sums up to the middle term; then
d. Write each factor in the pairs as the last term of the binomial factors.
Note: always perform factoring using common monomial factor first before applying any
type of factoring.
Assessment
Direction: Factor completely the following trinomials.
1. 2
x −5 x−24
2. 2
x +8 x−65
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3. x +9 x+ 20