Trinomials where A does not equal 1

What does that mean?
A trinomial in standard form is represented as y = ax2 + bx + cTo qualify as a trinomial the expression needs
A quadratic term (ax2) A linear term (bx) A constant (c)
If a (the coefficient in front of x2) is greater or less than 1 try....
How to solve the trinomial where A does not equal one

First try to solve the trinomial by

Factoring out the Greatest Common Factoror Solve using the Perfect Square Trinomial Method

If neither of these apply we can use the Reverse FOIL Methods demonstrated. There are two different options.

Option 1: Guess and Check
Works best when A is a prime number
We will use y=3x2+25x+28 as an example
  1. Find the first two terms of the two binomials
(3x+ ) (x+ )
Because 3x and x are the only two terms whose product is 3x2
2. Because all terms are positive the sign are all addition signs
3. Find two factors of 28 which, when FOILed complete the expression
2 and 14 are factors of 28 How to FOIL foil.jpg


Put the numbers into the expression


(3x+2)(x+14)


Find the middle term by multiplying the Inners and outer of FOIL

3x*14 and 2*x



Add them together


2x+42x=44x
44x is not the middle term we are looking for so we need to try again

4 and 7 are factors of 28

Put the numbers into the expression


(3x+4)(x+7)

Find the middle term by multiplying the Inners and outer of FOIL

3x*7 and 4*x


Add them together

21x+4x=25x

This was the term we were looking for, so the answer is,

(3x+4) (x+7)

Additional Example

2x2 -4x -6
Set up two binomials
(2x )(x )
Find factors of -6
2 and -3
Put the numbers into the expression
(2x-3)(x+2)
Find the middle term by multiplying the Inners and outer of FOIL

2x*2 and -3*x

Add them together

-3x+4x=x


That's not what we are looking for,
Try again, rearrange the numbers, put them into the equation

(2x+2)(x-3)
Find the middle term by multiplying the Inners and outers of FOIL
2x*-3 and 2*x
Add them together
2x+-6x=-4x

Everything works! (2x+2)(x-3)


Option 2: Factor by grouping
Works best when A is not a prime number
We will use 4x2+16x+5 as an example
  1. Multiply a and c
4*15=60
  1. Find all factors of 60 and their sums
Factors
1,60
2,30
3,20
4,15
5,12
6,10
Sums
61
32
23
19
17
16

  1. Look for the numbers whose sum equals the coefficient of the middle term
10 and 6 work
  1. Divide the middle term by the two factors
4x2+6x+10x+15
  1. Separate the first two and last two terms.
(4x2+6x) (10x+15)
  1. Factor out GCF
2x(2x+3) 5(2x+3)
  1. Rewrite the outside numbers as a second binomial

(2x+3) (2x+5)



Additional Problem

Factor 8x2+15x+7
  1. Multiply a and c
8*7=56
  1. Find all factors of 56 and their sums
Factors
1,56
2,28
4,14
7,8
Sums
57
30
18
15

  1. Look for the numbers whose sum equals the coefficient of the middle term
7 and 8 work
  1. Divide the middle term by the two factors
8x2+8x+7x+7
  1. Separate the first two and last two terms.
(8x2+8x) (7x+7)
  1. Factor out GCF
8x(x+1) 7(x+1)
  1. Rewrite the outside numbers as a second binomial
(8x+7) (x+1)

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