What is factoring by grouping? When factoring a trinomial by grouping, why is it necessary to write the trinomial in four terms?
math help needed.
In this method you are basically reversing the steps of multiplying a binomial times a binomial by FOIL
Let me illustrate with an example
expand (2x+3)(5x-4)
= 10x^2 - 8x + 15x - 12
notice that in the original expansion there are four terms
= 10x^2 + 7x - 12
Suppose you were given 10x^2 + 7x - 12 to factor. (of course you would not see the above expansion)
the trick is to separate the 7x into its original two components,
so the middle term will be re-written as the sum of two terms
which means you now have four terms.
I assume you know the actual method of proceeding from here, and you just needed an explanation why there are four terms.
Factoring by grouping is a technique used to factor a polynomial into its linear factors. It is particularly useful when dealing with a trinomial that does not have a common factor among all its terms.
To factor a trinomial by grouping, you typically write the trinomial as a sum of four terms. This is necessary because we are going to group the terms in pairs in order to factor out a common factor from each pair separately.
Let's consider an example to understand it better. Suppose we have the trinomial:
ax^2 + bx + cx + d
To factor this trinomial by grouping, we first group the terms:
(ax^2 + bx) + (cx + d)
Now, we focus on the first group and find the greatest common factor (GCF) of the terms in this group. Let's say the GCF is 'x'. By factoring out 'x', we get:
x(a + b)
Next, we move on to the second group and find the GCF of the terms in this group. Let's say the GCF is '1'. By factoring out '1', we get:
1(c + d)
Combining these two terms, we have:
x(a + b) + 1(c + d)
Finally, we can factor out the common factor 'x' from the first term and the common factor '1' from the second term, resulting in the factored form:
x(a + b) + 1(c + d) = (a + b)x + (c + d)
So, by writing the trinomial as four terms and grouping the terms, we are able to factor out a common factor from each pair separately, which eventually leads to factoring the trinomial.
In summary, writing a trinomial in four terms is necessary for factoring by grouping as it allows us to group the terms and factor out common factors from each pair separately.