Question

How do you factor $2{x^2} + 4x + 2$?

Answer 1

Hint: Here we need to factorize the given polynomial $2{x^2} + 4x + 2$. Note that the degree of the polynomial is 2, so it is a quadratic polynomial. Firstly, we note that 2 is common in the polynomial given. So we take it out and we obtain a perfect square trinomial. To solve this we make use of a perfect square trinomial rule which is given as ${(a + b)^2} = {a^2} + 2ab + {b^2}$. Substitute the values of the variables a and b and then solve to obtain the required result.

Complete step by step solution:
Given the polynomial of the form,
$2{x^2} + 4x + 2$ …… (1)
We are asked to factorize the above polynomial given in the equation (1).
Note that the degree of the above polynomial is 2. Hence it is a quadratic polynomial.
If we carefully observe the polynomial, the number 2 is common in all the terms.
So we take it out and solve further.
Taking out 2 in the equation (1), we get,
$ \Rightarrow 2({x^2} + 2x + 1)$ …… (2)
Observe that we have obtained a perfect square trinomial now. We solve it further to obtain the factors.
Now consider the polynomial, ${x^2} + 2x + 1$
This can also be written as,
$ \Rightarrow {x^2} + 2x + {1^2}$ …… (3)
Note that the above polynomial is of the form of a perfect square trinomial.
So we make use of the perfect square trinomial rule to solve the polynomial.
The perfect square trinomial rule is given by,
${(a + b)^2} = {a^2} + 2ab + {b^2}$
Note that here $a = x$ and $b = 1$.
Now we check the middle term by multiplying a and b by 2 and this results with the middle term in the original equation.
$2ab = 2 \cdot x \cdot 1$
$ \Rightarrow 2ab = 2x$
Hence the perfect square trinomial rule is satisfied.
Therefore, the equation (3) becomes,
$ \Rightarrow {(x + 1)^2}$
Substituting this in the equation (2), we get,
$ \Rightarrow 2{(x + 1)^2}$

Hence the factorization of the equation $2{x^2} + 4x + 2$ is given by $2{(x + 1)^2}$.

Note :
Firstly, take out the common term if it exists in the given polynomial, so that it makes our simplification easier. Students must know the perfect squares of the numbers. Otherwise it becomes difficult to solve this kind of problem.
Also they must know the perfect square trinomial rule to find the factors of a given equation.
We recognise the perfect square trinomial rule by the following facts.
(1) It contains three terms.
(2) Two of its terms are perfect squares themselves.
(3) The remaining term will be twice the product of the square root of the other two terms.
The perfect square trinomial formulas are given by,
${(a + b)^2} = {a^2} + 2ab + {b^2}$ and ${(a - b)^2} = {a^2} - 2ab + {b^2}$