Question

How do you factor \[6{{x}^{2}}+13x-5\]?

Answer 1

Hint: This type of problem is based on the concept of factoring a polynomial. First, we have to consider the polynomial with degree 2. We have to split the middle term of the polynomial in such a way that we get common terms from the first and last term. Here, the middle term is 13x. We can split the middle as an addition of 15x and 2x. We have to take 3x common from the first two terms and -1 common from the last two terms. We find that (2x+5) is common in the two terms. On taking the common terms, we get the factors of the polynomial.

Complete step by step solution:
According to the question, we are asked to find the factors of \[6{{x}^{2}}+13x-5\].
We have been given the polynomial is \[6{{x}^{2}}+13x-5\]. -----(1)
The given polynomial is of degree 2 and variable x.
To find the factors of the polynomial, we have to consider the middle term and split the middle term in such a way that the addition of these terms will be 13 and multiplication will be \[6\times -5\] which is equal to -30.
We know that 15-2=13 and \[15\times -2=-30\].
Therefore, we get
\[6{{x}^{2}}+13x-5=6{{x}^{2}}+\left( 15-2 \right)x-5\]
Using the distributive property, that is \[a\left( b+c \right)=ab+ac\].
Here, a=x, b=15 and c=-2.
\[\Rightarrow 6{{x}^{2}}+13x-5=6{{x}^{2}}+15x+\left( -2 \right)x-5\]
\[\Rightarrow 6{{x}^{2}}+13x-5=6{{x}^{2}}+15x-2x-5\]
We can express the polynomial as
\[\Rightarrow 6{{x}^{2}}+13x-5=3\times 2{{x}^{2}}+3\times 5x-2x-5\]
We find that 3x are common in the first two terms of the simplified polynomial and -1 are common in the last two terms of the simplified polynomial.
Let us take 3x and -1 common out of the bracket respectively.
\[\Rightarrow 6{{x}^{2}}+13x-5=3x\left( 2x+5 \right)-1\left( 2x+5 \right)\]
We find that 2x+5 is common in both the terms of the equation. On taking (2x+5) common, we get
\[6{{x}^{2}}+13x-5=\left( 2x+5 \right)\left( 3x-1 \right)\]
Here, we find that the polynomial is converted as a product of two linear polynomials which are the factors of the given polynomial.
Therefore, the factors of \[6{{x}^{2}}+13x-5\] are 3x-1 and 2x+5.

Note: Whenever you get this type of problem, we should always try to make the necessary changes in the middle term of the polynomial to get the required answer. Avoid calculation mistakes based on sign conventions. Since the polynomial is of degree 2, we get two factors.