in mathematics, factorization or factoring is the decomposition of an object (for example, a number...
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In mathematics, factorization or factoring is the decomposition of an object (for example, a number or a polynomial) into a product of other objects, or factors, which when multiplied together give the
original.
For example, the number 15 factors into primes as 3 × 5
In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something
to "basic building blocks”
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To factor an expression means to:
re-write it as a product
Why factor?
Once an expression has been factored, equivalent factors can divide to one
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Remember factor trees?
9
3 3X
9 1X
9 = 3 X 3
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Notice:
4(x + 2) = 4x + 8
Expanding
Factoring
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Common Factoring
A common factor can be divided out of every term in the polynomial
1. Number: GCF2. Variable: Common Variable, take
the lowest exponent
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Common Factor:
10x + 20 = 10( )
25x2 + 100x = 25x( )
4x + 12x2 – 16x3 =
4x(1 + 3x – 4x2)
8a2b3 + 12a4b2 =
4a2b2(2b + 3a2)
10x + 2
25 xx + 4
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YOU MAY DIVIDE OUT COMMON FACTORS
YOU MAY NOT “CROSS OUT” OR “CANCEL” SIMILAR TERMS
Consider the following example:
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3x + 63
, x = 2Direct
= 3(2) + 63
= 123
= 4
Factoring3x + 6
3= 3(x + 2)
3
1
1
= x + 2
= 4
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Crossing out
3x + 63
= x + 6
= 8
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Stop here…
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Trinomials: ax2 + bx + c
Simple (a = 1):
Add to the middle
Multiply to the last
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Factorx2 + 5x + 6 = (x )(x ) + 2 + 3
simple
Add: 5 Multiply: 6
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Factorx2 - 2x - 15 = (x )(x ) - 5 + 3
simple
Add: -2 Multiply: -15
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Complex: (a > 1) Decomposition
Add to the middle, multiply to (first)(last)
Common Factor twice
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Factor
6x2 – 1x – 2 = 6x2 – 4x + 3x – 2
= 2x(3x – 2) + 1(3x – 2)
= (3x – 2)(2x + 1)
You may also guess and check
complexAdd: -1 Multiply: -12CF the first pair, then CF the second pair
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Difference of Squares
x2 – y2 = (x – y)(x + y)
x2 – 25 = (x – 5)(x + 5)
4x2 – 36y2= (2x – 6y)(2x + 6y)
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Grouping
Sometimes, part of a 4 term polynomial can be “grouped” together and factored
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Factor
ax + cx + ay + cy
= x(a + c) + y(a + c)
= (a + c)(x + y)
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Factor
a2 – p2 + 2a + 1
= a2 + 2a + 1 – p2
=(a + 1)2– p2
=(a + 1 - p)(a + 1 + p)
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The Great Factoring Plan!!!
C.F. 3+
3
2
D of SSimple
Complex
Grouping
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