Question

How do you factor${a^3} + 3{a^2} - a - 3$?

Answer 1

Hint:To order to determine the factors of the above cubic equation ,compare the given equation with the standard cubic equation $A{x^3} + B{x^2} + Cx + D$ ,now take ${a^2}$ common from first two terms and -1 from the last two terms, use of the formula of$\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)$to find all the factors of the given cubic expression.
Formula:
$\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)$
$\left( {{A^3} - {B^3}} \right) = \left( {A - B} \right)\left( {{A^2} + A.B + {B^2}} \right)$
\[{A^2} - 2AB + {B^2} = {(A - B)^2}\]

Complete step by step solution:
Given a Cubic equation ${a^3} + 3{a^2} - a + 3$, let it be $f(x)$
$f(x) = {a^3} + 3{a^2} - a - 3$
Comparing the equation with the standard cubic equation $A{x^3} + B{x^2} + Cx + D$
A becomes 1
B becomes 3
C becomes -1
and D becomes -3
To find the cubic factorization,
Taking ${a^2}$ common from first two terms and -1 from the last two terms, we get
$
f(x) = {a^2}(a + 3) - 1(a + 3) \\
= (a + 3)({a^2} - 1) \\
$
applying formula $\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)$in the second factor by considering $A\,as\,a$ and $B\,as\,1$, our equation becomes
$ = \left( {a + 3} \right)\left( {a - 1} \right)\left( {a + 1} \right)$
Hence, we have successfully factorized our cubic equation.
Therefore, the factors are $\left( {a + 3} \right)\left( {a - 1} \right)\left( {a + 1} \right)$.
Additional Information:
Cubic Equation: A cubic equation is a equation which can be represented in the form of $a{x^3} + b{x^2}cx + d$ where $x$ is the unknown variable and a,b,c,d are the numbers known where $a \ne0$.
If $a = 0$ then the equation will become a quadratic equation and will no longer be cubic.
The degree of the quadratic equation is of the order 3.
Every Cubic equation has 3 roots.
The Graph of any cubic polynomial is symmetric with respect to the inflection point of the polynomial.
Graph to cubic polynomial ${a^3} + 3{a^2} - a - 3$

The points at which the graph touches the x-axis are the roots of the polynomial.

Note:
1. One must be careful while calculating the answer as calculation error may come.
2.Don’t forget to compare the given cubic equation with the standard one every time.