SOLUTION: Identify all the rational zeros for f(x)=x3(cubed)+4x2(squared)-3x-18

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Question 7756: Identify all the rational zeros for f(x)=x3(cubed)+4x2(squared)-3x-18
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Find the zeros of:
f%28x%29+=+x%5E3+%2B+4x%5E2+-+3x+-18 Try factor by grouping:
f%28x%29+=+%28x%5E3+-+8%29+%2B+%284x%5E2+-+3x+-+10%29 Take the first set of parentheses:
x%5E3+-+8 is a difference of two cubes and can be factored as follows:
%28A%5E3+-+B%5E3%29+=+%28A+-+B%29%28A%5E2+%2B+AB+%2B+B%5E2%29 where: A = x and B = 2, then:
%28x%5E3+-+8%29+=+%28x+-+2%29%28x%5E2+%2B+2x+%2B+4%29
Now the second set of parentheses:
4x%5E2+-+3x+-+10 Factor:
4x%5E2+-+3x+-+10+=+%284x+%2B+5%29%28x+-+2%29 So, altogether we have:

Now, factor out (x - 2):
f%28x%29+=+%28x+-+2%29%28x%5E2+%2B+2x+%2B+4+%2B+4x+%2B+5%29 Simplifying the 2nd parentheses:
f%28x%29+=+%28x+-+2%29%28x%5E2+%2B+6x+%2B+9%29 Now factor the 2nd parentheses:
f%28x%29+=+%28x+-+2%29%28x+%2B+3%29%5E2 Now, set this equal to zero.
%28x+-+2%29%28x+%2B+3%29%5E2+=+0 and apply the zero product principle:
x+-+2+=+0, then; x+=+2 or
%28x+%2B+3%29%5E2+=+0 then: x+%2B+3+=+0, and x+=+-3
So, the three zeros are:
x = 2, or x = -3 (a double root)
Take a look at the graph.
graph%28300%2C200%2C-5%2C5%2C-10%2C10%2Cx%5E3%2B4x%5E2-3x-18%29