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Question: How do you factor \({x^3} - 9{x^2} + 27x - 27\)?...

How do you factor x3−9x2+27x−27{x^3} - 9{x^2} + 27x - 27?

Explanation

Solution

To order to determine the factors of the above cubic equation ,use of the formula of(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)for the first and last term and pull common −9x - 9xfrom both left terms .We’ll get a product to linear and quadratic equation and to factorise the quadratic one use the formula A2−2AB+B2=(A−B)2{A^2} - 2AB + {B^2} = {(A - B)^2}to find all the factors .

Complete step by step solution:
Given a Cubic equationx3−9x2+27x−27{x^3} - 9{x^2} + 27x - 27,let it be f(x)f(x)
f(x)=x3−9x2+27x−27f(x) = {x^3} - 9{x^2} + 27x - 27

Comparing the equation with the standard cubic equation ax3+bx2cx+da{x^3} + b{x^2}cx + d
a becomes 1

b becomes -9

c becomes 27

and d becomes -27

To find the cubic factorization,

First rearranging the terms,
f(x)=x3−27−9x2+27x f(x)=x3−33−9x2+27x  f(x) = {x^3} - 27 - 9{x^2} + 27x \\\ f(x) = {x^3} - {3^3} - 9{x^2} + 27x \\\
Now applying formula(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)in the first two terms taking A as xA\,as\,xand B as 3B\,as\,3 pulling out common −9x - 9xfrom the last two terms
=(x−3)(x2+3x+9)−9x(x−3)= (x - 3)({x^2} + 3x + 9) - 9x(x - 3)

Taking common (x−3)(x - 3)
=(x−3)(x2+3x+9−9x)= (x - 3)({x^2} + 3x + 9 - 9x)

Combining all like terms
=(x−3)(x2−6x+9) =(x−3)(x2−2(3)(1)x+32)  = (x - 3)({x^2} - 6x + 9) \\\ = (x - 3)({x^2} - 2(3)(1)x + {3^2}) \\\

The quadratic part of the expression can be factored using formulaA2−2AB+B2=(A−B)2{A^2} - 2AB + {B^2} = {(A - B)^2}

Now our equation becomes
=(x−3)(x−3)2= (x - 3){(x - 3)^2}
Using property of exponentam×an=am+n{a^m} \times {a^n} = {a^{m + n}}
=(x−3)3= {(x - 3)^3}
f(x)=(x−3)3f(x) = {(x - 3)^3}
Hence, we have successfully factorized our cubic equation.

Therefore, the factors are (x−3)(x−3)(x−3)=(x−3)3(x - 3)(x - 3)(x - 3) = {(x - 3)^3}

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}

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 + dwhere xxis the unknown variable and a,b,c,d are the numbers known where a≠0a \ne 0.If a=0a = 0then 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 x3−9x2+27x−27{x^3} - 9{x^2} + 27x - 27

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.