Download Factoring, The Fun Never Ends

Factoring, The Fun Never Ends
Rule 1: Take out all common factors – ALWAYS DO THIS FIRST (The Distributive Property)
3xy
3x2y – 6xy2
What’s common? ______
3x2y – 6xy2 = 3xy (x – 2y)
2x2
What’s common? ______
2x4 – 8x2
5x + 10x2 – 40x3
2x4 – 8x2 = 2x2 (x2 – 4)
5x
What’s common? ______
5x + 10x2 – 40x3 = 5x (1 + 2x – 8x2)
Rule 2: For TWO TERMS ( factor using the rules below or it does not factor)
*if quadratic use difference of squares: x2 – y2 = (x – y)(x + y)
4 – a2b2 = (2 – ab) (2 + ab)
9x2 – y2 = (3x – y) ( 3x + y )
16c2 – 25a2 = ( 4c – 5a ) (4c + 5a )
1 – 36x2y2 = ( 1 – 6xy) ( 1+ 6xy )
*if cubic use difference of cubes: x3 – y3 = (x – y)(x2 + xy + y2)
or sum of cubes: x3 + y3 = (x + y)(x2 – xy + y2)
8 – a3 = ( 2 – a ) (4 + 2a + a2)
x3 + 8a3b3 = ( x + 2ab) (x2 – 2abx + 4a2b)2
27x3 – 1 = ( 3x – 1) (9x2 + 3x + 1 )
8y3 + 125 = ( 2y + 5) (4y2 – 10y + 25 )
Notice: the trinomial terms come from the binomial terms, first term = binomial first term
squared, middle term = product of both with change of sign, and last term = last squared.
Rule 3: For THREE TERMS, Perfect Square Trinomials and General Strategy
*Perfect Square Trinomial formula:
16 – 8x + x2 = (4 – x)(4 – x) = (4 – x)2
x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2
x2 – 2xy + y2 = (x – y) (x – y) = (x – y)2
4x2 + 12xy + 9y2 = (2x + 3y)(2x + 3y) = (2x+ 3y)2
Factoring Rules continued
Rule 3: For THREE TERMS – *General Strategy – explained using the following example
Example: 4x2 – 7x – 15
Step 1- multiply outside numbers 4 and 15 getting 60
4·15
Step 2- form all the two number products that equal 60
2·30
Step 3- select the product pair whose sum or difference is
12·5 difference is 7
equal to the middle term coefficient in the given problem.
20·3
(sum if 4 and 15 have the same sign, difference otherwise)
6·10
1· 60
Step 4- write the number pair as terms with x and with
signs so that adding them gives the middle term, –7x : -12x + 5x = -7x
Step 5- replace the –7x in the original with the two terms that equal it and group in pairs
4x2 – 12x + 5x – 15 = (4x2 – 12x) + (5x – 15) when possible put the negative term first
Step 6- factor each pair using the distributive property: 4x (x – 3) + 5 (x – 3)
now use the distributive once more getting the factors: (x – 3) (4x + 5)
So factoring the trinomial 4x2 – 7x – 15 = (x – 3) (4x + 5)
Rule 4: For FOUR TERMS – Group in pairs (steps 5 & 6 above) or group 3 terms & one as below
9 – a2 + 2ab – b2 = 9 – (a2 – 2ab + b2) notice how a parenthesis after a negative sign changes signs
= 9 – (a – b)2 and using difference of squares = [3 – (a – b)][3 + (a – b)]
So 9 – a2 + 2ab – b2 = (3 – a + b) (3 + a – b)
More examples, first we’ll finish the last two examples given with Rule 1:
* 2x4 – 8x2 = 2x2 (x2 – 4) from Rule 1, but since the last factor is two terms and fits the difference
of squares (Rule 2) the completely factored form is: 2x4 – 8x2 = 2x2 (x – 2) (x + 2)
* 5x + 10x2 – 40x3 = 5x (1 + 2x – 8x2) from Rule 1, but since the last factor has three terms Rule
3 may be applied: 1 times 8 is 8, factors of 8 that have a difference of 2 are 4 and 2, -2x + 4x =
2x so 1 + 2x – 8x2 = 1 – 2x + 4x – 8x2 = (1 – 2x) + (4x – 8x2) = 1(1 – 2x) + 4x(1 – 2x),
therefore 5x + 10x2 – 40x3 = 5x (1 – 2x) (1 + 4x)
Notice the above two examples required more than one rule to factor completely, the procedure
for factoring then is FIRST apply Rule 1, SECOND count the terms and use the correct rule to
finish. There are some polynomials that cannot be factored more, for example all the trinomials
in the examples for the sum or difference of cubes formulas, i.e. 9x2 + 3x + 1. Can you tell why?
* 8x + 27x4 = x (8 + 27x3) = x (2 + 3x) (4 – 6x + 9x2) What rules were used? _________
Rules 1 & 2
sum of cubes
Rule 4
* 2ax + 6a – bx – 3b = (2ax + 6a) – (bx + 3b) What rule is being used ______
inserted after minus sign
Why is the negative in front of 3 changed to +? Parenthesis
_____________________
distributive
= 2a (x + 3) – b (x + 3) = (x + 3) (2a – b) what property is used twice? _______
* 25x2 + 70x + 49 = (5x + 7) (5x + 7) what rule? _____,
Rule 3 the answer can also be written? (5x
+ 7)2
______
Both cubic rules
* a6 – b6 = (a3 + b3) (a3 – b3) what rule? ______,
Rule 2 which other rules can be applied? ________
So a6 – b6 = (a + b) (a2 – ab + b2) (a – b) (a2 + ab + b2)