Question 1043460
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Let f(x)=x^4+2x^3-7x^2-20x-12
If -2 is a zero, factor f(x) completely 
- please show how to factor it as well. 
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{{{graph( 330, 330, -5.5, 5.5, -5.5, 5.5,
          x^4+2x^3-7x^2-20x-12
)}}}


<B>Figure</B>. Plot f(x) = {{{x^4+2x^3-7x^2-20x-12}}}

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<pre>
Let's apply grouping method.

f(x) = {{{x^4+2x^3-7x^2-20x-12}}} = {{{(x^4+2x^3) - (7x^2 +20x +12)}}} = 

= {{{(x^3(x+2))}}} - {{{(7x^2 + 14x) + (6x+12)}}} = 

= {{{(x^3(x+2))}}} - {{{(7x(x + 2)) + 6(x+2)}}} = 

= {{{(x+2)*(x^3 - 7x -6)}}} =       (now check the polynomial {{{x^3 -7x -6}}} for the roots. Check for x =1. 
                             See how I will extract this root by applying grouping again)


= {{{(x+2)*((x^3 -x) - (6x+6))}}} = {{{(x+2)*(x(x^2-1) - 6(x+1))}}} = {{{(x+2)*(x(x-1)*(x+1) - 6(x+1))}}} =

= {{{(x+2)*(x+1)*(x*(x-1)-6)}}} = = {{{(x+2)*(x-1)*(x^2 - x-6)}}} =

     Now you have to find the roots of the quadratic polynomial in parentheses.
     You can do it by many ways. Its roots are 2 and -3. Therefore

=  {{{(x+2)*(x-1)*(x+2)*(x-3)}}} =    (so you finally get)

=  {{{(x+2)^2(x-1)*(x-3)}}}.

Now identify the roots of the polynomial.
Check the correspondence between the roots and the plot in the figure.


What method did I apply?  - The grouping method.

What method did I apply?  - The grouping method.
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See also the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/81-Solving-polynomial-equations-of-high-degree-by-factoring.lesson>Solving polynomial equations of high degree by factoring</A>

in this site.