creating brackets. in this powerpoint, we meet 5 different methods of factorising. type 1 – common...
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Creating brackets
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In this powerpoint, we meet 5 different methods of factorising.
Type 1 – Common Factor
Type 2 – Difference of Two Squares
Type 3 – Grouping
This involves taking a term outside the brackets. Always try to do this first.Try this when you have two terms with a minus between
This is the easiest one to pick – use it when there are 4 terms!
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Types 4 and 5
Quadratic trinomials
Use these for expressions with 3 terms.
They will be of the format
x2 + bx + c (Type 4) OR
ax2 + bx + c (Type 5)
Where a, b and c are just numbers
Factorising just makes me sooooo happy!!
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Summary
TypeType When to UseWhen to Use
1. Common 1. Common factorfactor
Always try first before any other methodAlways try first before any other method Examples: Examples: aa22 – 9 – 9aa ; 2 ; 2xy xy + 5+ 5xx22
2. Difference 2. Difference of Two of Two squaressquares
When there are only When there are only 2 terms2 terms which are which are squaressquares There must be a There must be a minus signminus sign Examples: Examples: aa22 – 25 ; 81 – 4 – 25 ; 81 – 4bb22 ; ; ww44 – 16 – 16
3. Grouping3. Grouping There are There are 4 terms.4 terms. Example: Example: aa22 – 4 – 4aa + 3 + 3ab – ab – 1212bb
4. Quadratic 4. Quadratic Trinomial (I)Trinomial (I)
There are There are 3 terms. Has a squared term.3 terms. Has a squared term. Examples: Examples: aa22 – 9 – 9aa + 20 ; 6 – 5 + 20 ; 6 – 5bb + + bb22
5. Quadratic 5. Quadratic Trinomial (II)Trinomial (II)
There are There are 3 terms. Has a squared term 3 terms. Has a squared term with a number attached in front.with a number attached in front. Examples: 2Examples: 2aa22 – 3 – 3aa – 5 ; 6 – 5 ; 6bb – 5 – 5bb22 + 3 + 3bb
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Type 1 of 5 – common factor
Always try this first, regardless of what type it is
3a – 12 = 3(a – 4)
3a2 – 12a =
3a2 + 6a + 12 =
20ab – 12b2 =
30a6 – 15a5 =
3a(a – 4)
4b(5a – 3b)
15a5(2a – 1)
3(a2 + 2a + 4)
Remember – take out the largest factor you can!
Always look for a
common factor!
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Type 2 of 5 – diff of 2 squares
To qualify as a Type 2, an expression
• must have only 2 terms which are SQUARES
• must have a MINUS sign separating them
Examples
a2 – 9 = (a – 3)(a + 3)
16 – a2 = (4 – a)(4 + a)
(2b)2 – (3a)2 =
9b2 – 25 = (3b – 5)(3b + 5)
(2b – 3a)(2b + 3a)
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Combining Types 1 and 2
Example 1 .....Factorise 5x2 – 45
STEP 1 Treat as a Type 1, and take out common factor first, 5Write 5(x2 – 9)
STEP 2 Now do expression in brackets as a Type 2
Write 5(x – 3)(x + 3)...ANS!
LookMum ! It’s a
difference of 2
squares!
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Example 2 .....Factorise x4 – 81
STEP 1 Treat as a Type 2, and write as difference of 2 squares.....(x2 – 9)(x2 + 9)
STEP 2
(x2 – 9)(x2 + 9)
(x – 3)(x + 3)(x2 + 9)....ANS!!
Now check out the thing in each bracket. We can factorise the first one, but not the second.
Y’can’t factorise a SUM of two squares Stupid! x2 + 9 has to stay as it is. It’s not
the same as (x + 3)(x + 3) is it now???
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Example 3 .....Factorise 80a4 – 405b12
STEP 2
STEP 3
STEP 1 Identify common factor, 5 and remove
Write 5(16a4 – 81b12)
Now work on the terms in the brackets
This is a difference of 2 squares and becomes (4a2 – 9b6) (4a2 + 9b6)
Now work on the terms in the 1st bracket.
This is a difference of 2 squares and becomes (2a – 3b3) (2a + 3b3) . Write
Write 5(4a2 – 9b6) (4a2 + 9b6)
5(2a – 3b3) (2a + 3b3) (4a2 + 9b6)
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Example 4 .....Factorise 9a2 – (x – 2a)2
Just treat as difference of 2 squares of the format
9a2 – b2 where the b = [x – 2a]
Factorising it then becomes
= (3a – b)(3a + b)And then replacing the b with [x – 2a] we get
= (3a – [x – 2a])(3a + [x – 2a])Now get rid of square brackets
= (3a – x + 2a)(3a + x – 2a)Clean up
= (5a – x )(a + x) Ans!!You could check your answer by expanding it and also expanding the original question. They should both give the same thing.
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Type 3 of 5 – Grouping You can tell when you’ve got one of these because
there are FOUR TERMS !!!
Example 1Factorise 2a – 4b + ax – 2bx
STEP 1 – split it into “2 by 2” = 2a – 4b + ax – 2bx
STEP 2 – factorise each pair separately as Type 1 = 2(a – 2b) + x(a – 2b)
STEP 3 – take out the (a – 2b) as a common factor
= (a – 2b)(2 + x)...ans!!
No need to be confused!
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Type 3 of 5 – Grouping
Example 2
Factorise xy + 5x – 2y – 10
STEP 1 – split it into “2 by 2” = xy + 5x – 2y – 10
STEP 2 – factorise each pair separately as Type 1 = x(y + 5) – 2 (y + 5)
STEP 3 – take out the (y + 5) as a factor
= (y + 5)(x – 2) ans!!
If these are the same, it’s a good
sign!
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Type 3 of 5 – Grouping
Example 3
Factorise x2 – x – 5x + 5
STEP 1 – split it into “2 by 2” = x2 – x – 5x + 5
STEP 2 – factorise each pair separately as Type 1 = x(x – 1) – 5 (x – 1)
STEP 3 – take out the (x – 1) as a factor
= (x – 1 )(x – 5) ans!!
Ewbewdy!!They’re the same! On my way to a VHA
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Example 4 - harder
Factorise x2 – 4y2 – 2ax – 4ay
STEP 1 – split it into “2 by 2”
= x2 – 4y2 – 2ax – 4ay
STEP 2 – factorise each pair separately
= (x – 2y) (x + 2y)
STEP 3 – take out the (x + 2y) as a factor
= (x + 2y)(x – 2y – 2a) ans!!
– 2a (x + 2y)
1st pair – Type 2
2nd pair – Type 1
Awwright! They’re the
same!!
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Type 4 of 5 – Easy Quadratic Trinomial
Example 1 .....Factorise x2 + 5x + 6
You can usually pick these as they have 3 TERMS
STEP 1 – Make 2 brackets
(x..............)(x.............)
STEP 2 – Look for 2 numbers that
Multiply to make +6
Add to make +5 +2 & +3
STEP 3 – Put ‘em in the brackets (x + 2)(x + 3)
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Type 4 of 5 – Easy Quadratic Trinomial
Example 2 .....Factorise 2x2 – 6x – 20 STEP 1 – take out a common factor (remember this should be your 1st step EVERY time!!)
= 2(x2 – 3x – 10)
STEP 2 – Ignore the 2. For the expression inside the brackets, look for 2 numbers that
Multiply to make – 10
Add to make – 3 +2 & – 5
STEP 3 – Put ‘em in the brackets2(x + 2)(x – 5)
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Type 4 of 5 – Easy Quadratic Trinomial
Example 3 .....Factorise 6 + 5x – x2 STEP 1 – Rearrange into “normal” format with x2 at the front, then x, then the number
= – x2 + 5x + 6
STEP 2 – Now take out a common factor – 1
STEP 3 – Ignore the minus. Look for 2 numbers that add to – 5, and multiply to – 6.
= – (x2 – 5x – 6)
These are +1 and –6. – (x + 1)(x – 6)
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Type 5 of 5 – Harder Quadratic Trinomial
Example 1 .....Factorise 2x2 + 5x – 3 STEP 1 – Draw up a fraction like this
2........)2........)(2( xx
STEP 2 – Look for two numbers that
ADD to make +5
MULT to make – 6
2 × – 3 = – 6
Numbers are +6, – 1 2
)12)(62(
xx
= (x + 3)(2x – 1) ANSNote the 2 in bottom must cancel one whole bracket FULLY! So (2x + 6) becomes (x + 3)
With a number in front of the x2
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Type 5 of 5 – Harder Quadratic Trinomial
Example 2 .....Factorise 3x2 + 8x – 3 STEP 1 – Draw up a fraction like this
3........)3........)(3( xx
STEP 2 – Look for two numbers that
ADD to make +8
MULT to make – 9
3 × – 3 = – 9
Numbers are +9, – 1 3
)13)(93(
xx
= (x + 3)(3x – 1) ANSNote the 3 in bottom must cancel one whole bracket FULLY! So (3x + 9) becomes (x + 3)
With a number in front of the x2
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Type 5 of 5 – Harder Quadratic Trinomial
Example 3 .....Factorise 6x2 – 19x + 10 STEP 1 – Draw up a fraction like this
6........)6........)(6( xx
STEP 2 – Look for two numbers that
ADD to make –19
MULT to make 60
6 × 10 = 60
Numbers are –4 , –15 32
)156)(46(
xx
= (3x – 2)(2x – 5) ANSNote the 6 in bottom would not cancel either bracket FULLY! So we broke the 6 into 2 x 3 then cancelled.
With a number in front of the x2
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Now wozn’t that just a barrel of fun??