Jessa Barber

Choose a factoring example, and follow the steps in order to

solve it. It can be 2, 3 or 4 terms.

Step 1.

How many terms are there?

2 3 4

Step 2.

Is there a GCF?

YES NO

Step 3.

If there is no GCF, then ‘factor by grouping’.Ex) pq + pr – sq – sr

a) Group the terms that have common variables (pq + pr) – (sq – sr)

b) Remove a GCF from

the grouped terms

p (q + r) – s (q + r)

d) * When factoring by grouping, the brackets must be the same, or you have done something wrong

p (q + r) – s (q + r)

c) Factor out the brackets by removing a GCF from the new set of terms.

(q + r) (p – s)

If the1st term that you are grouping is negative, make the whole bracket

negative.

Click, if the brackets are not the same.

NEXT

Step 3.

If there is no GCF, then ‘factor by grouping’.Ex) pq + pr – sq – sr

a) Group the terms that have common variables (pq + pr) – (sq – sr)

b) Remove a GCF from

the grouped terms

p (q + r) – s (q + r)

d) * When factoring by grouping, the brackets must be the same, or you have done something wrong

p (q + r) – s (q + r)

c) Factor out the brackets by removing a GCF from the new set of terms.

(q + r) (p – s)

If the1st term that you are grouping is negative, make the whole bracket

negative.

Click, if the brackets are not the same.

NEXT

If brackets are not the same:

Ex) 20x(x - 3) – 4(3 - x) The brackets are similar, but not exactly the same.

a) Make brackets the same, by factoring out a (-1)

Ex) 20x(x - 3) – (- 1) 4(x - 3)

= 20x(x-3) + 4(x-3)

Go back

Step 2.

Is there a GCF?

YES NO

Step 2.

Is there a GCF?

YES NO

Step 3.

Which form is the trinomial in?

x2 + bx + c

ax2 + bx + c or ax2 + bxy + cy2

Where ‘a’ does notequal 1

x2 + bxy + cy2

Step 4.‘Factoring Quadratic Trinomials’

Where the leading coefficient is not 1, and does not factor out

There are 3 methods which allow you to solve:

“OI” Decomposition Rumsy’s

Choose the one you find most helpful:

Step 5.

“Oi Method” – Guess and check Check as you go, but only the inside and outside. FOIL

1. Ex) 2x2 + 7x + 6 – Nothing factors out

2. Draw 2 sets of brackets – ( )( )

3. Find 2 numbers that multiply to 1st term – (2x )(x )

4. Guess and check to find two numbers that multiply to the last term, but also work with FOIL

(2x + 3) (x + 2) 4x + 3x = 7x

3 x 2 = 6

3x

4x NEXT

DecompositionEx) 2x2 + 7x + 6

a) Multiply 1st number to last number 2 x 6 = 12

b) Find two numbers that multiply to number in step a, and add to middle term of trinomial. 3 x 4 = 12 3 + 4 = 7

c) Replace the middle term with the two numbers from step b.

2x2 + 4x + 3x + 6

d) Factor our a GCF in pairs (factor by grouping)

= (2x2 + 4x) + (3x + 6)

= 2x(x + 2) + 3(x + 2)

e) Factor out brackets

(x + 2) (2x + 3)

Step 5.

NEXT

Rumsy’s MethodEx) 2x2 + 7x + 6• Multiply 1st number to last number 2 x 6 = 12• Replace last number with number in step a. 2x2 + 7x + 12• Draw 2 sets of brackets, and put the first term as well as the first variable (x) on

the left of each bracket. (2x ) (2x )• Find the last 2 numbers by finding two numbers that multiply to equal the last term,

and add to equal the middle term (2x + 3)(2x + 4)

3 x 4 = 123 + 4 = 6

e) Divide one or both sets of brackets by a GCF but do not put the GCF outside of but do not put the GCF outside of the brackets.the brackets. (2x + 3)(x + 2)

Step 5.

NEXT

Step 4.‘Factoring Quadratic Trinomials 2’ – (x2 + bxy + cy2)

Ex) n2 + 10ny + 21y2

• Draw 2 sets of brackets( )( )

2. The 1st term in each bracket must multiply to get the 1st term in the trinomial.(n )(n )

4. Find two numbers that multiply to the last term, and add to the second. Because the last term has a variable2, you must add a variable to these terms:7y x 3y = 21y2 and 7y + 3y = 10y2

5. Put those on the right of the brackets (n + 3y)(n + 7y) NEXT

Step 4.

‘Factoring Quadratic Trinomials’ – (x2 + bx + c)Ex) n2 + 10n + 21• Draw 2 sets of brackets

( )( )

2. The 1st term in each bracket must multiply to get the 1st term in the trinomial.(n )(n )

3. Find two numbers that multiply to the last term, and add to the second. 7 x 3 = 21 7 + 3 = 10

4. Put those on the right of the brackets (n + 3)(n + 7)

NEXT

Step 3.

Is it a difference of squares?

Ex) 4x-16 (To be a difference of squares, it must be a perfect square – perfect squareperfect square – perfect square

YES NO

Step 4.How to factor a difference of squares:

Ex) 9x2-16y2

a) Draw 2 sets of brackets ( )( )

b) The first term in both sets of brackets, must multiply multiply to get the first term.

(3x )(3x ) Because it is a perfect square, the first numbers will be equal to each other.

c) The second term in each bracket must also multiply to get the the second term.second term.

(3x 4y)(3x 4y)

d) Put a positive sign in one bracket, and a negative sign in the other. (This is so that once it is FOILed out, the second term will still be negative)

(3x + 4y)(3x – 4y)NEXT

Step 4.

Does it look something like this?

- 4x(x+7) – 3 (x+7)

a) Remove the GCF. In this case, (x+7) is the GCF

b) Divide each term by the GCF - (x+7)

= (x + 7)(4x – 3)

Click, if the brackets are not

the same.

NEXT

Step 5.

If brackets are not the same:

Ex) 20x(x-3) – 4(3-x) The brackets are similar, but not

exactly the same.

a) Make brackets the same, by factoring out a (-1)

Ex) 20x(x-3) – (- 1) 4(x-3) = 20x(x-3) + 4(x-3)

NEXT

Step 6.

Does it look something like this? - 20x(x-3) + 4(x-3)

a) Remove the GCF. In this case, (x-3) is the GCF

b) Divide each term by (x-3)

= (x-3)(20x + 4) This can be simplified

c) To simplify, divide each term in the second bracket by 4, and then put the 4 outside of the brackets.

= 4(x-3)(5x+1)

NEXT

Step 3.

Finding and removing a GCF: The GCF between terms uses only the common common

basesbases and lowest exponents.lowest exponents.

Example:Example: 12ab, 15a2b3 GCF = 3ab

Find the GCF, and put it outside of a set of brackets. GCF( )

Then divide each term inside the bracketseach term inside the brackets with the GCF

NEXT

Step 4.

Now, is your example fully factored?

Ex) Factored: 3ab(x+1)

Not Fully Factored: 3ab(x2+y2)

Factored Not Factored

Step 5.

Now, is your example fully factored?

Ex) Factored: 3ab(x+1)

Not Fully Factored: 3ab(x2+y2)

Factored Not Factored

Step 4.

Check that you have factored correctly by using ‘FOIL’

Step 5/6.

Check that you have factored correctly by using ‘FOIL’

Step 7.

Check that you have factored correctly by using ‘FOIL’

DONE!