Short notes on Factoring
EMT111
Laurel Benn
September 23, 2010
Factoring an expression means writing it as a product.
One of the easiest tools to use in factoring is common factors. To use this we
would take out all the factors that are common to every term in the expression.
Example 1 Factor 4x3 y 4 − 16xy 3 + 20x3 y 2
We see that all 3 terms contains a factor of 4xy 2 . So taking out this common
factor we get 4x3 y 4 − 16xy 3 + 20x3 y 2 = 4xy 2 (x2 y 2 − 4y + 5x2 )
Special Formulas
Another good tool to use is special formulas.
Here are some special formulas that we should know:
1. x2 − y 2 = (x + y)(x − y)
2. x3 + y 3 = (x + y)(x2 − xy + y 2 )
3. x3 − y 3 = (x − y)(x2 + xy + y 2 )
4. x2 + (a + b)x + ab = (x + a)(x + b)
5. x2 + 2xy + y 2 = (x + y)2
Now let’s look at some examples on the usage of these formulas.
Example 2 (using x2 − y 2 = (x + y)(x − y))
1. w2 − 9 = (w + 3)(w − 3)
2. 16x2 − 9y 2 = ((4x)2 − (3y)2 ) = (4x + 3y)(4x − 3y)
√ √ √ √ √ √
3. 3t2 − 2 = ( 3t)2 − ( 2)2 = ( 3t + 2)( 3t − 2)
4. x4 − y 4 = (x2 + y 2 )(x2 − y 2 ) = (x2 + y 2 )(x + y)(x − y)
Example 3 (using x3 + y 3 = (x + y)(x2 − xy + y 2 ))
1. 27x3 + 8y 3 = (3x)3 + (2y)3 = (3x + 2y)(9x2 − 6xy + 4y 2 )
2. 8a3 + 1 = (2a)3 + 13 = (2a + 1)(4a2 − 2a + 1)
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Example 4 (using x2 + (a + b)x + ab = (x + a)(x + b))
1. x2 + 6x + 8 = (x + 2)(x + 4)
2. t2 − 9t + 14 = (t − 2)(t − 7)
Grouping
This method can be helpful at times.
Example 5 Factor 15ab + 12b + 10a + 8 Here we see that the first 2 terms
have 3b as a common factor while the last 2 terms have 2 as a common factor.
Making the first 2 terms one “Group” and the last 2 terms another “Group”,
we factor the groups separately to get: 3b(5a + 4) and 2(5a + 4). This means
that 15ab + 12b + 10a + 8 = 3b(5a + 4) + 2(5a + 4) and 5a + 4 is a common
factor. Taking out this factor we get (5a + 4)(3b + 2)
⇒ 15ab + 12b + 10a + 8 = (5a + 4)(3b + 2)
Example 6 Factor 6ax + 3ay − 4bx − 2by + 10x + 5y.
With this let’s group the first two terms, then the second two, then the third two.
We should then get the following:
6ax+3ay−4bx−2by+10x+5y = 3a(2x+y)+(−2b)(2x+y)+5(2x+y) = (2x+y)(3a−2b+5)
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