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How do I find the roots, solutions, zeros, or what have you, of 2x 3 +13x 2 +23x+12 without graphing it and observing the points where y=0?

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How do I find the roots, solutions, zeros, or what have you, of 2x3+13x2+23x+12 without graphing it and observing the points where y=0?

Jan 23, 2018

#1
+100572
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2x^3+13x^2+23x+12  =  0

Let's see if we can find an alternative way to write this.

A little trial and error produces  a way to break up 13x^2   as  2x^2 + 11x^2

So we have

2x^3  + 2x^2  + 11x^2 + 23x + 12      factor by grouping

2x^2 ( x + 1)  +  (11x + 12) ( x + 1)        the common factor is  x + 1

So we have

(x + 1)  [  2x^2 + 11x + 12 ]  =  0        factor the second polynomial

( x + 1)  ( 2x  + 3) ( x + 4)  =  0

Setting each factor to o and solving for x  produces

x + 1  = 0             2x + 3  = 0           x  + 4  = 0

x = -1                    2x  =  -3              x  = -4

x  = -3/2

The  zeroes are in red

BTW  - the first factoring trick won't always work...but....it is something to try

Jan 24, 2018