Question 1155519
a) Find all zeros of the polynomial algebraically 

{{{ P(x)=18x^4-21x^3-81x^2+84x+36 }}}

b) Then write the polynomial in factored form.

{{{P(x)=18x^4-21x^3-81x^2+84x+36}}} 

{{{P(x) = 3 (6 x^4 - 7 x^3 - 27 x^2 + 28 x + 12)}}}

{{{P(x)=3  (6 x^4 + 5 x^3 +17 x^2 - 6x-12x^3-10x^2-34x+12) }}}

{{{P(x)=3  ((6 x^4 -12x^3)+ (5 x^3-10x^2) + (17 x^2-34x) - (6x-12)) }}}

{{{P(x)=3  (6x^3(x -2)+ 5x^2(x-2) + 17x( x-2) -6 (x-2)) }}}

{{{P(x)=3 (x - 2)(6 x^3 + 5 x^2 - 17 x - 6) }}}

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

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

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

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

{{{P(x)=3 (x - 2) ( x +2 )(6x^2+2x-9x-3)}}}

{{{P(x)=3 (x - 2) ( x +2 )((6x^2+2x)-(9x+3))}}}

{{{P(x)=3 (x - 2) ( x +2 )(2x(3x+1)-3(3x+1))}}}

{{{P(x)=3 (x - 2) (x + 2) (2x - 3) (3x + 1) }}}

zeros:

if {{{(x - 2)=0}}}=>{{{x=2}}}
if {{{(x + 2)=0}}}=>{{{x=-2}}}
if {{{(2x - 3)=0}}}=>{{{x=3/2}}}
if {{{(3x + 1)=0}}}=>{{{x=-1/3}}}




c) Sketch the graph of P(x) showing all real zeros, y-intercept, and end behavior. 


{{{ graph( 600, 600, -5, 5, -50, 50, 3 (x - 2) (x + 2) (2x - 3) (3x + 1)) }}}