What is factoring by grouping? When factoring a trinomial by grouping, why is it necessary to write the trinomial in four terms?
product of two binomials will become a trinomial.This is bcz itwo terms of the product are like terms so you can simplify and write them as oe term.When you need to factorize you need to have 4 terms.That is why you split the middle term of the trinomial to get a quadrinomial, then group two terms to get a common factor.
See the example
x2 -6x +9 = 0
split the middle term
x2 -3x -3x +9 =0 now make two groups
x2 -3x -3x +9 =0
x(x-3) -3(x-3) = 0 now take (x-3)
(x-3)(x-3) = 0
)(x-3) = 0
10x+15y=465
x+y=39
x-?
y-?
factor by grouping 20^3+65s^2-60s
Factoring by grouping is a method used to factorize a trinomial expression that involves grouping terms together. It is called "factoring by grouping" because we group the terms in a specific way to find common factors.
To understand why it is necessary to write the trinomial in four terms, let's take a closer look. A trinomial typically has the form ax^2 + bx + c, where a, b, and c are constants and x is the variable. In order to factorize it using the grouping method, we need to write it as the sum or difference of two binomials.
The key idea behind factoring by grouping is to group the terms in a way that allows us to find common factors. By writing the trinomial in four terms, we can usually group the first two terms together and the last two terms together.
For example, let's consider the trinomial 2x^2 + 3x - 5. To factorize it, we can write it as (2x^2 + 3x) + (-5). By grouping the terms correctly, we can now factor out the greatest common factor from each group of terms separately.
In this case, the greatest common factor in the first group, (2x^2 + 3x), is x. Factoring out x gives us x(2x + 3). Similarly, in the second group, the greatest common factor is -1, so factoring out -1 gives us -1(5).
Now, we have two binomials, x(2x + 3) - 1(5), and we can further simplify it to (2x + 3)x - 5, which is the factored form of the trinomial.
So, writing the trinomial in four terms and grouping them allows us to identify common factors within each group, making it easier to factorize the trinomial using the method of factoring by grouping.