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1. Factor completely the polynomial P(x)=x^3 + 9x^2 + 24x + 20 using synthetic division. Show work.  

I BELIEVE the answer is - p(x)=(x+2)2(x+5) 
But I could be wrong. any thoughts?? 

2. Use one of the factors & the original P(x), show the factoring while using the long division of polynomials. Only 1 division of polynomials is required. 
^^^ this is where i get confused. help?

 

 
tylersomers2000  Mar 26, 2015

Best Answer 

 #1
avatar+90023 
+13

1.   P(x)=x^3 + 9x^2 + 24x + 20

By the Rational Roots Theorem, -2 is one root...so we have

-2     1    9      24      20

             -2     -14    -20

----------------------------

        1    7      10      0

The remaining polynomial  is x^2 + 7x + 10    which factors as (x + 2) (x + 5)

So the factorization of this is    P(x)  =  (x +2)(x + 2) (x +5)  = (x + 2)^2 (x + 5)

 

2.  We have

                            x^2   + 7x   + 10

             --------------------------------------------

x + 2    [  x^3  + 9x^2  + 24x  + 20 ]  

                x^3  + 2x^2

             -------------------------------------------

                           7x^2  + 24x

                            7x^2 + 14x

             --------------------------------------------

                                        10x  + 20

                                         10x + 20

             ------------------------------------------

                                                     0  remainder

So, now notice, tyler, the quotient polynomial,  x^2 + 7x + 10,   factors as before  in the first part

So the three factors  are (x + 2)(x + 2)(x + 5)  =  (x + 2) ^2 (x + 5)  .....as before....

 

  

CPhill  Mar 26, 2015
 #1
avatar+90023 
+13
Best Answer

1.   P(x)=x^3 + 9x^2 + 24x + 20

By the Rational Roots Theorem, -2 is one root...so we have

-2     1    9      24      20

             -2     -14    -20

----------------------------

        1    7      10      0

The remaining polynomial  is x^2 + 7x + 10    which factors as (x + 2) (x + 5)

So the factorization of this is    P(x)  =  (x +2)(x + 2) (x +5)  = (x + 2)^2 (x + 5)

 

2.  We have

                            x^2   + 7x   + 10

             --------------------------------------------

x + 2    [  x^3  + 9x^2  + 24x  + 20 ]  

                x^3  + 2x^2

             -------------------------------------------

                           7x^2  + 24x

                            7x^2 + 14x

             --------------------------------------------

                                        10x  + 20

                                         10x + 20

             ------------------------------------------

                                                     0  remainder

So, now notice, tyler, the quotient polynomial,  x^2 + 7x + 10,   factors as before  in the first part

So the three factors  are (x + 2)(x + 2)(x + 5)  =  (x + 2) ^2 (x + 5)  .....as before....

 

  

CPhill  Mar 26, 2015
 #2
avatar+93677 
0

That was a lot of effort Chris,  looks really good :)

Melody  Mar 27, 2015
 #3
avatar+90023 
0

Thanks, Melody.....

 

  

CPhill  Mar 27, 2015

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