Question

How do you factor: $2{x^2} + 7x + 3?$

Answer 1

Hint:For a polynomial of the form $a{x^2} + bx + c$ , rewrite the middle term as a sum of two terms whose product is $a \times c$ and whose sum is $b$.

Complete step by step answer:
We have to find the factors of $2 \times 3 = 6$ which sum to $ + 7$ are $ + 6$ and $ + 1$
Split the middle term using these factors
$ \Rightarrow 2{x^2} + 6x + x + 3$ (factor by grouping)
$ = 2x\left( {x + 3} \right) + 1\left( {x + 3} \right)$
Take out the common factor $\left( {x + 3} \right)$
$ = \left( {x + 3} \right)\left( {2x + 1} \right)$
$ \Rightarrow 2{x^2} + 7x + 3$
$ = \left( {x + 3} \right)\left( {2x + 1} \right)$

Note: We can factorize a given polynomial using the split middle term method. Always try to split the middle term such that the product of first and last terms is equal to the product of split terms.
Let’s go through the following steps:
First step: Multiply the coefficient of ${x^2}$ with the constant term. We get $2 \times 3 = 6$.
We can split $7$ as $6 + 1,5 + 2,4 + 3$ and the first and last term multiplies into $6\left( {3 \times 2} \right)$.
Second step: Split the above product into two factors. Such factors are $6,1;2,3;$.
We can split $7x$ as $6x + x$
Third step: Choose that set of factors whose sum gives you the coefficient of the $x$ term. They are $6$ and $1$.
Fourth step: Write the given expression as $2{x^2} + 6x + x + 3$
$ = 2x\left( {x + 3} \right) + 1\left( {x + 3} \right)$
$ = \left( {x + 3} \right)\left( {2x + 1} \right)$.