Question

How do you factor $f(x) = {x^4} + 6{x^3} + 9{x^2}?$

Answer 1

Hint: In this problem we use the concept of factorization. Note that the degree of the above equation is 4. So firstly, we convert the equation into the quadratic equation which is of degree 2. We do this by taking out ${x^2}$ throughout the equation. This converts the quartic equation into quadratic. Then we try to factorize the quadratic equation using the basic algebraic formula given by ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.

Complete step by step solution:
Given the equation $f(x) = {x^4} + 6{x^3} + 9{x^2}$
 Note that the degree of the above equation is 4. Hence it is an example of a quartic equation.
We will try to convert this quartic equation into quadratic equation.
Quadratic equation is an equation whose highest degree is 2.
We note that the term ${x^2}$ is common throughout the equation.
So to convert $f(x)$ into a quadratic equation we take out ${x^2}$ from the entire given equation and we obtain a quadratic equation. We then try to factorize it.
We have ${x^4} + 6{x^3} + 9{x^2}$
Taking out ${x^2}$ in the above equation we get,
$ \Rightarrow $ ${x^2}\left( {{x^2} + 6x + 9} \right)$ ……(1)
Now we simplify the quadratic equation to factorize it.
Consider the quadratic equation $\left( {{x^2} + 6x + 9} \right)$
This quadratic equation can also be written as,
${x^2} + 6x + 9 = {x^2} + 2.3.x + {3^2}$
Now we factorize the quadratic equation using some basic algebraic formula.
Note that the above equation is of the form ${a^2} + 2ab + {b^2}$, where a, b, c are any real numbers.
We have the algebraic formula ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ ……(2)
We now make use of this formula to factorize the quadratic equation.
Note that in the equation (1), $a = x$, $b = 3$
Substituting the values of a and b in the formula given in the equation (2), we get,
$ \Rightarrow {x^2} + 2.3.x + {3^2} = {\left( {x + 3} \right)^2}$
Hence substituting this in the equation (1) we get,
${x^2}\left( {{x^2} + 6x + 9} \right) = {x^2}{\left( {x + 3} \right)^2}$

Hence the factorization of $f(x) = {x^4} + 6{x^3} + 9{x^2}$ is ${x^2}{\left( {x + 3} \right)^2}$.

Note:
Factorization is a process in which a number or an expression is written in the forms of its smaller factors which on multiplication gives the original number or expression.
We have to make sure that the given polynomial is quadratic or not.
If it is not, we try to convert into quadratic by taking out common terms.
We make use of basic algebraic formulas which are important to solve these kinds of problems.
The formulas which we need to remember are given below.
(1) ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$
(2) ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$