Question

What is the new AC method to factor trinomials?

Answer 1

Hint: In this problem we need to write the method used to factorize the trinomials. We know that the trinomial is a polynomial of having three non-zero terms. So, we will first assume a trinomial and write the method to factorize it.

Complete step-by-step solution:
Let us assume the trinomial as $a{{x}^{2}}+bx+c$.
In the above polynomial we can observe that there are only three non-zero terms. So, we can treat it as a trinomial.
Now factoring the trinomial $a{{x}^{2}}+bx+c$ is carried out by first calculating the product of the coefficients $a$ and $c$. So, we will calculate the value of $ac$.
After getting the value of $ac$, we will list all the factors for the calculated value of $ac$. In the list of factors, we need to choose two factors such that their algebraic sum should be equal to $b$. That means if we consider any two factors, say $p$ and $q$, then they must satisfy the below conditions.
One is $p+q=b$ and the another one is $pq=ac$.
Now we will split the term $bx$ which is in $a{{x}^{2}}+bx+c$ as $px+qx$. So, the polynomial is modified as
$a{{x}^{2}}+bx+c=a{{x}^{2}}+px+qx+c$
Now we will take appropriate terms as common from the set of two terms in the above equation and simplify the equation to get the factors of the trinomial.

Note: In this method we have only discussed the trinomial which is in the form of $a{{x}^{2}}+bx+c$. There are different types of trinomials like $a{{x}^{3}}+b{{x}^{2}}+cx$, $a{{x}^{5}}+b{{x}^{2}}+c$ and so on. For this type of trinomials just write each and every coefficient in its factorization form and take appropriate terms as common include the variables that means $x$ in this case and simplify the equation to get the factors.