Question

How do you factor $6{{x}^{2}}+5x+1$?

Answer 1

Hint: In this question, we need to find the factors of the expression $6{{x}^{2}}+5x+1$. For this, we will use to split the middle term method. In this, we will compare the equation with $a{{x}^{2}}+bx+c$ then we need to find two numbers such that their product is equal to the product of a and c and also their sum is equal to b. After finding them we will substitute them for b and split it. Then we will take common factors from first two terms and common factors from the last two terms. Using this, we will be able to factorize the expression.

Complete step by step answer:
Here we are given the expression as $6{{x}^{2}}+5x+1$. We need to factorize it. For this let us use the split the middle term method.
For using this method, let us first compare the given equation with $a{{x}^{2}}+bx+c$ to find the value of a, b and c. Comparing $6{{x}^{2}}+5x+1$ with $a{{x}^{2}}+bx+c$ we get a = 6, b = 5 and c = 1.
Now we need to split the middle term.
Let us first take two numbers such that their sum is equal to b and their product is equal to the product of a and c. If these two numbers are ${{n}_{1}},{{n}_{2}}$ we need to find them such that,
${{n}_{1}}\cdot {{n}_{2}}=a\cdot c=6\cdot 1=6$ and ${{n}_{1}}+{{n}_{2}}=b=5$.
Factors for 6 are 1, 2, 3, 6. The only two numbers whose sum is 5 and multiplied together give product as 6 are 2 and 3. So we will split 5 into 2+3. Our expression become, $6{{x}^{2}}+\left( 2+3 \right)x+1$.
Opening bracket we get $6{{x}^{2}}+2x+3x+1$.
Now let us take 2 as common factor from first two terms and 1 as common factor from last two terms we get $2x\left( 3x+1 \right)+1\left( 3x+1 \right)$.
We see that, we have two terms with common factor (3x+1) so taking it out we get $\left( 3x+1 \right)\left( 2x+1 \right)$.
Hence our expression has been factorized.
The two factors in which the expression is factorized are (2x+1) and (3x+1).
So (2x+1) and (3x+1) are the required factors.

Note:
Students should always try to make factors of a.c terms so that they can easily find factors, which add to become b. Make sure to take common terms such that we are left with the same factor inside both brackets (as in (3x+1)). Do not try to take 6-1 = 5 as it will not be multiplied to get 6. It will give -6. So take care of signs also.