Beginning Algebra

5.3 More Trinomials to Factor

BeginningAlgebra

6.3 More Trinomials to Factor

Objective 1. To factor a trinomial whose leading coefficient is other than 1.

Objective 2. To factor a polynomial by first factoring out the greatest common factor and then factoring the

polynomial that remains.

Read as + or –

Same sign + Sum

Ax2 Bx C

Larger sign – Difference

Clue of Signs

A C = P P = r s, r > s

r + s = B

oror

r - s = B- Difference

+ Sum

Ax2 Bx C

Larger sign

Same sign

Ax2 Bx C

Grouping NumberGrouping Number GN = P

1. Find the productproduct of the firstfirstand last coefficientslast coefficients: AA and CC

2. Find all of the pairs of factorspairs of factors rr and ss

Given a general quadratic trinomialgeneral quadratic trinomial:

What to do How to do it

is the middle coefficient .r - s = Br - s = B

orr + s = Br + s = B3. so their sumsum

P = rP = r ss, r > sr > s

Ax2 Bx C

AA CC = P

Factor general quadratic trinomial:

or differencedifference

is the middle coefficient .

Factor by Clue of SignsClue of Signs: What to Do How to Do It

Ax2 Bx C

[Read as “+ “+ oror ”” ]

Given general trinomial of type that has no common factor. Read the clues of the signsclues of the signs.

The product P = AP = ACC is the grouping numbergrouping number

(r ‑s) = Bwhose sum or difference is B

+ sum

Find all possible factors of GN = PP = rs , r > s

(r + s) = B

difference

Rewrite middle termmiddle term BxBx: Ax2 rx rx sx sx + C

and factor by groupingfactor by grouping (ax b)(cx d)

Sum

+

Difference

Same signSame sign

Larger signLarger sign

2. Find all of the pairs of factors: r and s

1. Find the productproduct of 1010 and 66:

3. Middle sign is + therefore:

60 = 60 · 1

10 · 6 = 60

+15 , - 4


15 - 4 =1115 · 4

12 · 510 · 6

20 · 330 · 2

4. Separate middle termmiddle term 11x11x: +15x , - 4x

with the difference difference = 11.

Example: 10x2 + 11x - 6

Example: 10x2 + 11x - 6

5. Copy the polynomial:

6. Rewrite middle termmiddle term 11x11x: and group for factoringgroup for factoring

7. Factor each groupeach group: bring down middle signmiddle sign

8. Factor common factorcommon factor:

10x2 + 15x - 4x+ 15x - 4x - 6

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

10x2 + 11x - 6


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

Check Factors using FOIL What to Do How to Do It

CheckCheck by multiplying back using

First

Note sum of O + I terms

F 0 I L

(5x - 2)(2x + 3)Outer

Inner

Last 10x2 + 15x - 4x - 6

10x2 - 6+15x

- 4x

10x2 ++ 11x 6= (5x 2)(2x ++ 3)

Example: 3x2 - 7x - 6



3. Middle sign - is larger signlarger sign:

18 = 18 · 1

3 · 6 = 18

- 9 , + 2


9 - 2 = 7 6 · 3

9 · 2

4. Separate middle termmiddle term - 7x- 7x: - 9x , + 2x



6. Rewrite middle termmiddle term -7x -7x:

and group for factoringgroup for factoring



3x2 - 9x- 9x + 2x+ 2x - 6

3x(x - 3) ++ 2(x - 3)

3x2 - 7x - 6


(3x + 2)(x - 3)

Example: 3x2 - 7x - 6



First


F 0 I L

(3x + 2)(x - 3)

Outer

Inner

Last 3x2 - 9x + 2x - 6

3x2 - 6 - 9x

+ 2x

3x2 -- 11x - 6= (3x + 2)(x -- 3)


1. Look at Look at numbers onlynumbers only Find the productproduct of 33 and 66:

3. Middle sign - is same signsame sign:

18 = 18 · 1

3 · 6 = 18

- 9 , - 2


9 + 2 = 11 6 · 3

9 · 2

4. Separate middle termmiddle term - 11xy- 11xy: - 9xy , - 2xy- 9xy , - 2xy

with the sumsum = 11.

Example: 3x2 - 11xy + 6y2


6. Rewrite middle termmiddle term -11xy-11xy: and group for factoringgroup for factoring



3x2 - 9xy- 9xy - 2xy- 2xy + 6y2

3x(x - 3y) - 2(x - 3y)

3x2 - 11xy + 6y2


(3x - 2y)(x - 3y)

Example: 3x2 - 11xy + 6y2



First


F 0 I L

(3x - 2y)(x - 3y)

Outer

Inner

Last 3x2 - 9xy - 2xy + 6y2

3x2 + 6y2- 9xy

- 2xy

3x2 -- 11xy + 6y2

= (3x -- 2)(x -- 3)

Example: 6t2 + 23t + 20



3. Middle sign ++ is same signsame sign:

120 · 1

GN: 6 ·20 = 120

+15 , + 8


15 + 8 = 23 30 · 4

24 · 520 · 6

40 · 3

60 · 2

4. Separate middle termmiddle term 23t23t: +15t , + 8t

with the sumsum = 23. 15 · 8

12 · 10

120

5. Copy the trinomial:

6. Rewrite middle termmiddle term 23t23t:

and group for factoringgroup for factoring



6t2 + 15t + 8t+ 15t + 8t - 20

3t(2t + 5) + 4(2t + 5)

6t2 + 23t + 20


(3t + 4)(2t + 5)

Example: 6t2 + 23t + 20



First


F 0 I L

(3t + 4)(2t + 5)Outer

Inner

Last 6t2 + 15t + 8t + 20

6t2 + 20+15t+ 8t

6t2 ++ 23t ++ 20= (3t ++ 4)(2t ++ 5)

What to Do How to Do It

Ax2 Bx C

k·(axax22 bx bx

cc)

k·ax2 k·bx

k·c

1. Factor out the common factor(ss) from each termeach term.

2. Apply the distributive propertydistributive property.

Trinomials with Common FactorsCommon Factors:

3. As common factorscommon factors numbers are left in composite formcomposite form andand letters are left in power formpower form.

4. Check Inner Polynomial for Clue of SignsClue of Signs and GN

axax22 bx bx

cc

What to Do How to Do It

12x2y - 33xy + 9y

3y(4x2 - 11x + 3)

3y·4x2 - 3y·11x + 3y·31. Factor out the common factor(ss) from each term.

2. Apply the distributive propertydistributive property.

Trinomials with Common FactorsCommon Factors:

4. Check Inner Polynomial for Clue of SignsClue of Signs and GN

4x2 - 11x + 3

3. As common factorscommon factors numbers are left in composite formcomposite form andand letters are left in power formpower form.

Inner Trinomial: 4x2 - 11x - 3



3. Middle sign – is larger signlarger sign:

12 = 12 · 1

4 · 3 = 12

- 12 , + 1


12 - 1 = 11 4 · 3

6 · 2

4. Separate middle termmiddle term - 11x- 11x: - 12x , + 1x


5. Copy the trinomial:

6. Rewrite middle termmiddle term --11x11x: and group for factoringgroup for factoring



4x2 - 12x- 12x + 1x+ 1x - 3

4x(x - 3) ++ 1(x - 3)

4x2 - 11x - 3


(4x + 1)(x - 3)

Inner Trinomial: 4x2 - 11x - 3

CompleteComplete: Multiply common factorcommon factor 3y 3y(4x + 1)(x - 3)

Check Factors by FOIL What to Do How to Do It

Check factors of inner trinomialinner trinomial by

First

Find the sum of O + I terms

F 0 I L

(4x + 1)(x - 3)

4x2 - 11x - 3

Outer

Inner

Last 4x2 - 12x + 1x - 3

4x2 - 3- 12x

+ 1x

Now, multiply by common factorcommon factor 3y3y 12x2y -- 33xy - 9y

= 3y(4x + 1)(x -- 3)

THEEND

5.3

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Beginning Algebra