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How do you factorize the trinomial $3{x^2} - 14x - 5$?

Last updated date: 13th Jun 2024
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Hint: Here we have to find out the roots of the given trinomial expression by the factoring method. On splitting the middle term of the given expression and doing some simplification we get the required answer.

Complete step-by-step solution:
Given polynomial is $3{x^2} - 14x - 5$
We need to find out the roots for the trinomial polynomial by using factoring.
First we need to put it is equal to zero
So, we can write it as $3{x^2} - 14x - 5 = 0$
Now solving the given trinomial polynomial as follows:
$3{x^2} - 14x - 5$
We need to find the combination of two numbers which when multiply each other it would be $ - 15$and $x$ when subtracted both the numbers, so we get
We can write the equation as,
 $ \Rightarrow 3{x^2} - 15x + x - 5 = 0$
Taking common the desired number from the above equations in half parts
$ \Rightarrow 3x(x - 5) + 1(x - 5) = 0$
Arranging the equation as we want as roots
$ \Rightarrow (x - 5)(3x + 1) = 0$
We put each bracket equal to zero
$ \Rightarrow x - 5 = 0$ and $3x + 1 = 0$
Hence we get
$ \Rightarrow x = 5$ and $x = - \dfrac{1}{3}$

Therefore, the roots for the given trinomial $3{x^2} - 14x - 5 = 0$ are \[5\] and $ - \dfrac{1}{3}$

Note: In algebra, we discuss in the primary level class; A polynomial contains three terms or we can say that any three monomials, is known as a trinomial polynomial, where monomial is the polynomial which has only one term.
The equation contains three variables that are $a,b{\text{ and c}}$. Therefore it is trinomial polynomial and Examples as $4{a^4} + 3x - 2,8{x^2} + 2x + 7$.
The equations contain only one element that is $z$ therefore, it is not a trinomial.
In solving these types of questions we need to understand the key concept of solving and we have to understand all the terms used in the question one by one.