Wize High School Algebra I Textbook (Common Core) > Factoring Polynomials

Factoring Numbers

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Prime Factorization
To write the prime factorization of a whole number, we need to rewrite that number into a product of prime numbers.

Examples
The prime factorization of 444 is    2  x  2   
The prime factorization of 101010 is    2  x  5   
This can get tricky when the number we want to find the prime factorization of is very large.

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Factor Tree
A factor tree is a tool that helps us find the prime factorization of a number visually.
At each level of the factor tree, we rewrite the previous number as a product of 2 numbers
We continue drawing more levels, stopping only when we've reached a prime number
The prime factorization of a number is the product of all of the prime numbers at the bottom of the factor tree

Example
Draw a factor tree to determine the prime factorization of 5400.

Therefore, the prime factorization of 5400 is 2×3×3×3×2×5×2×52\times 3\times3\times3\times2\times5\times2\times52×3×3×3×2×5×2×5.

Rewriting this using exponents, we see that the prime factorization is 23×33×52\boxed{2^3\times3^3\times5^2}23×33×52​.

Practice: Prime Factorization
Write the prime factorization of 6930.

Enter your answer in the form

a^x\times b^y\times c^z...

I don't know

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The Greatest Common Factor (GCF)
The greatest common factor of two numbers is the largest number that "goes into" both numbers.

Example
The GCF of 42 and 70 is 14, because 14 is the largest number that goes into both 42 (42÷14=342\div14=342÷14=3) and 70 (70÷14=570\div14=570÷14=5).

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How to Find the GCF?
Method 1
List out all factors of both numbers (numbers that go into both numbers), then find the largest one -- that's the GCF.

Example
The factors of 42 are      1, 2, 3, 6, 7, 14, 21, 42     

1×422×213×146×7\begin{array}{c}
1\times42\\
2\times21\\
3\times14\\
6\times7
\end{array}1×422×213×146×7​

The factors of 70 are      1, 2, 5, 7, 10, 14, 35, 70     

1×702×355×147×10\begin{array}{c}
1\times70\\
2\times35\\
5\times14\\
7\times 10
\end{array}1×702×355×147×10​

The largest number that both lists of factors have in common in    14   
, so the GCF is    14   
.

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Method 2
Write out the prime factorization of both numbers, then write down everything that they have in common -- that's the GCF.

Example
The prime factorization of 42 is    2 x 3 x 7   

The prime factorization of 70 is    2 x 5 x 7   

The two factorizations have    2 x 7   
 in common, so the GCF is    14   

Practice: Greatest Common Factor (GCF)
Determine the GCF of the following numbers:

a) 756 and 1575
[write out the all factors for both numbers to find the GCF]

b) 3300 and 2520 
[write the prime factorizations for both numbers to find the GCF]

a) The GCF of 756 and 1575 is

b) The GCF of 3300 and 2520 is

I don't know

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The Idea behind Common Factoring
Let's say you have the following 3 pencil cases:


It's clear that there are items in common (the same) between all 3 pencil cases. How can we simplify this picture using numbers?

Wize Tip
When thinking about common factoring, you want to "take out" (divide out) the largest common item as possible!

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Factoring with Numbers
Given any number nnn, its factors are divisors of nnn (meaning numbers that "go into nnn very nicely").

*Note: We usually only consider integer factors!

Example 1
Given the number 24, its factors are -1, 1, -2, 2, -3, 3, -4, 4, -6, 6, -8, 8, -12, 12, -24, and 24.

So, we can rewrite 24 as follows
24=24=24=     -1 (-24)     
24=24=24=     1(24)     
24=24=24=     -2 (-12)     
24=24=24=     2 (12)     
24=24=24=     -3 (-8)     
24=24=24=     3 (8)     
...

Example 2
Identify the greatest common factor between the following terms, then rewrite the expression by dividing out the greatest common factor.

a) 5+25=5+25=5+25=          5 (1 + 5)           


b) 6−15=6-15=6−15=          3 (2 - 5)           or          -3 (-2 + 5)          


c) 8−20=8-20=8−20=          4 (-2 - 5)           or          -4 (-2 + 5)               


d) −8−20=-8-20=−8−20=          -4 (2 + 5)           