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Question: How do you factor\({a^3} + 3{a^2} - a - 3\)?...

How do you factora3+3a2−a−3{a^3} + 3{a^2} - a - 3?

Explanation

Solution

To order to determine the factors of the above cubic equation ,compare the given equation with the standard cubic equation Ax3+Bx2+Cx+DA{x^3} + B{x^2} + Cx + D ,now take a2{a^2} common from first two terms and -1 from the last two terms, use of the formula of(A2−B2)=(A−B)(A+B)\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:
(A2−B2)=(A−B)(A+B)\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)
(A3−B3)=(A−B)(A2+A.B+B2)\left( {{A^3} - {B^3}} \right) = \left( {A - B} \right)\left( {{A^2} + A.B + {B^2}} \right)
A2−2AB+B2=(A−B)2{A^2} - 2AB + {B^2} = {(A - B)^2}

Complete step by step solution:
Given a Cubic equation a3+3a2−a+3{a^3} + 3{a^2} - a + 3, let it be f(x)f(x)
f(x)=a3+3a2−a−3f(x) = {a^3} + 3{a^2} - a - 3
Comparing the equation with the standard cubic equation Ax3+Bx2+Cx+DA{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 a2{a^2} common from first two terms and -1 from the last two terms, we get
f(x)=a2(a+3)−1(a+3) =(a+3)(a2−1)  f(x) = {a^2}(a + 3) - 1(a + 3) \\\ = (a + 3)({a^2} - 1) \\\
applying formula (A2−B2)=(A−B)(A+B)\left( {{A^2} - {B^2}} \right) = \left( {A - B} \right)\left( {A + B} \right)in the second factor by considering A as aA\,as\,a and B as 1B\,as\,1, our equation becomes
=(a+3)(a−1)(a+1)= \left( {a + 3} \right)\left( {a - 1} \right)\left( {a + 1} \right)
Hence, we have successfully factorized our cubic equation.
Therefore, the factors are (a+3)(a−1)(a+1)\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 ax3+bx2cx+da{x^3} + b{x^2}cx + d where xx is the unknown variable and a,b,c,d are the numbers known where a≠0a \ne0.
If a=0a = 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 a3+3a2−a−3{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.