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NOTABLE PRODUCTS AND FACTORIZATIONS - Coggle Diagram
NOTABLE PRODUCTS AND FACTORIZATIONS
Notable products
It is the multiplication betwee 2 or more minominals
Resulting in:
Trinomial
Polynomial
They are multiplied by themselves
Binomial
Binomial squared
The sum of two terms squared
It is equal to:
The square of the first term, plus twice the product of the first term by the second term, plus the square of the second term.
In more visual terms:
(a+b)2= a2 +2ab + b2
Product of conjugated binominals
Two binomials are conjugated when the second terms of each are of different signs, that is, that of the first is positive and that of the second negative or vice versa
Each notable product is a formula that results from a factorization, composed of factors.
The factors are the basis of a power and have an exponent
When the factors multiply, the exponents must be added.
Factorizations
It is when you split an entity into its factors or divisors
Example:
Factorization of the number 12 might look like 3 times 4
You can break that down even further using prime factorization, when you reduce a number to prime factors
Example:
12 can be factorized to 3 times 2 times 2
An entity can be a number, a matrix, or a polynomial
In the factorisation method, we reduce any algebraic or quadratic equation into its simpler form.
The factors of any equation can be:
Integer
Variable
Algebraic expression
Factorization Methods
Regrouping terms method
Means rearranging the given expression based on the like terms or similar terms.
For example, 2xy + 3x + 2y + 3 can be rearranged as:
2xy + 3x + 2y + 3
Expanding the terms into factor form.
= 2 × x × y + 3 × x + 2 × y + 3
Rearrange to get the common factor
= x × (2y + 3) + 1 × (2y + 3)
Now (2y + 3) is the common factor we can take out.
= (2y + 3) (x + 1)
Factorisation using identities
By using the common identities, we can factorise the given expression
Example: Factorise 4x2 – 9.
Solution: By using the algebraic identities, we know;
a2 – b2 = (a – b) (a+b)
Hence, we can write: 4x2 – 9= (2x)2 – 32 = (2x + 3) (2x – 3)
Note: For the rest identities check with the formulas for factorisation mentioned before.
Common factors method
We take out the common factors among each term of the given expression.
Example: Factorise 3x + 9.
3x + 9 = 3(x+3)
Factors of the form (x+a) (x+b)
If a given expression is in the form of x2 + (a + b) x + ab, then the factors will be (x+a) and (x + b).
