Question

Simplify the polynomial $\left( {{x^2} + 6x + 9} \right)$ using suitable algebraic identity.

Answer 1

Hint: Given polynomial is of degree 2. Polynomials of degree 2 are known as Quadratic polynomials. Quadratic polynomials can be factorised using various methods like the splitting the middle term, completing the square method and using algebraic identities. The given polynomial $\left( {{x^2} + 6x + 9} \right)$ can be factorized using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ and the expression can be condensed as a whole square.

Complete step by step answer:
Given question requires us to simplify the polynomial $\left( {{x^2} + 6x + 9} \right)$ using algebraic identities.
So, let the polynomial $\left( {{x^2} + 6x + 9} \right)$ be $p(x)$.
Then, $p(x) = \left( {{x^2} + 6x + 9} \right)$
We can see that the given expression somewhat resembles the identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
Thus, we can use algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ to condense the given expression as the whole square of a binomial.
In order to use the algebraic identity, we should convert the given polynomial into a form that would resemble the algebraic identity.
$p(x) = {x^2} + 2\left( 3 \right)\left( x \right) + {3^2}$
Now, using identity, ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$, we get,
$ \Rightarrow p(x) = {\left( {x + 3} \right)^2}$

So, the factorized and simplified form of the given polynomial is $p(x) = {\left( {x + 3} \right)^2}$.

Note: Besides using algebraic identities, there are various methods to simplify the quadratic expressions such as completing the square method, splitting the middle term and using the Quadratic formula. Splitting of the middle term can be a tedious process at times when the product of the constant term and coefficient of ${x^2}$ is a large number with a large number of divisors. Special care should be taken in such cases.