Question

Factorize the equation given below:
${x^3} + 3{x^2} - x - 3$

Answer 1

Hint: In order to solve this question, we will use basic identities of algebra such as ${a^2} - {b^2} = (a + b)(a - b)$ . First we will take out the common terms and try to see if any identity is applicable, we will use that identity and simplify it further.
Complete step-by-step answer:
Let $f(x) = {x^3} + 3{x^2} - x - 3$
For simplifying a given function of x, we will proceed further by taking common terms.
So, $f(x) = {x^2}(x + 3) - 1(x + 3)$
Now, $(x + 3)$ is common in both the term, so taking out $(x + 3)$
$f(x) = ({x^2} - 1)(x + 3)$
We know that $\left[ {{a^2} - {b^2} = (a + b)(a - b)} \right]$
By using the given identity, we get
$f(x) = (x + 1)(x - 1)(x + 3)$
Hence, the factors of ${x^3} + 3{x^2} - x - 3$ are $(x + 1),(x - 1)\& (x + 3)$

Note: In order to solve these types of questions, remember the basic formulas and properties of algebra. There are four basic properties of algebra such as commutative, associative, distributive and identity. You should be familiar with each of these. It is especially important to understand these properties in order to answer questions related to algebraic equations.