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Factoring:
Multiply factors to find a product.
When we break down a product into its factors, that’s called factoring.
Prime Number & Composite Number
A prime number (or a prime) is a Natural Number greater than 1 that has no positive divisors other than 1 and itself.
Example, 5 is Prime because 1 and 5 are its only positive integer factors
A natural number greater than 1 that is not a prime number is called a COMPOSITE NUMBER.
Example, 6 is composite because it has the Divisors by 2 and 3 in addition to 1 and 6.
Hint: every whole number is either Prime or composite except the number 1, which is considered neither prime nor composite.
Prime Factorization
"Prime Factorization" is finding which prime numbers multiply together to make the original number.
If it is not divisible, then it is a prime number such as 1, 2, 3, 5, 7 because these have one 1 number and itself. Such as (1, 2), (1,3), (1, 5) (1, 9)
The Term 8x can be factored into (8)(x) = 8 / 4 = 2 ((4* 2) = 8) 4, & 2 is the factor of 8
The Term 3x^{2} can be factored into (3)(x^{2}) = 3 is a prime number, so it cannot be factored any further, but x^{2} is equal to (x)(x). 3x^{2} = (3)(x* x)
But if you know your Special Binomial Products you might see it as the "difference of squares":
Because 4x^{2} is (2x)^{2}, and 9 is (3)^{2},
so we have:
4x^{2}  9 = (2x)^{ 2}  (3)^{2}
Hint: To check that you’ve factored correctly, multiply your factors. You should get back your original expression.
Colon is factor:
Example: 18x^{3} + 12x
First, find the largest whole number that divides evenly into both coefficients.
The factors of 18: (what can be multiplied – 1, 2, 3, 6, 9, and18
The factors of 12: (what can be multiplied – 1, 2, 3, 4, 6, and12
(Coefficient) (Base is x) (Exponent is 3)
6x: 18x^{3} / 6x = 3x^{2} (because of the division x is minus by 1)
6x: 12x / 6x = 2
Answer: 6x(3x^{2} + 2) = 6x(3x^{2}) + 6x(2) = 18x^{3} + 12x