SOLUTION: Identify the degree of the polynomial. Find all zeros (real and imaginary) and indicate their multiplicity if greater than 1. {{{P(x)=(x - 8)^3(x^2 +49)(4x^2 - 12x - 1)}}}

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: Identify the degree of the polynomial. Find all zeros (real and imaginary) and indicate their multiplicity if greater than 1. {{{P(x)=(x - 8)^3(x^2 +49)(4x^2 - 12x - 1)}}}      Log On

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Question 1156238: Identify the degree of the polynomial. Find all zeros (real and imaginary) and indicate their multiplicity if greater than 1.
P%28x%29=%28x++-++8%29%5E3%28x%5E2+%2B49%29%284x%5E2++-++12x++-++1%29
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
There is no need to multiply that out, collect terms and write in descending
order.  But I went ahead and did it anyway, and got:

4x%5E7-108x%5E6%2B1251x%5E5-9620x%5E4%2B57647x%5E3-211560x%5E2%2B291648x+%2B+25088

So we see that the degree is 7, the largest exponent of x.

However, we could have told that by observing that the factor (x-8)³ would
contribute a term in x³, the factor (x²+49) would contribute a term in x²
and the factor (4x²-12x-1) would contribute a term in x², so we add the
powers 3+2+2=7 and know that the degree is 7 without multiplying it out.

To find all the zeros, we set the right side of P(x) equal to zero and use the
zero-factor property:

P%28x%29=%28x++-++8%29%5E3%28x%5E2+%2B49%29%284x%5E2++-++12x++-++1%29 

%28x%5E%22%22++-++8%29%5E3%28x%5E2+%2B49%29%284x%5E2++-++12x++-++1%29=0

(x-8)³ = 0; x²+49 = 0;   4x²-12x-1 = 0
   x-8 = 0;    x² = 49;
     x = 8;     x = ±7i;

The 8 has multiplicity 3.
That's because if we wrote 
(x-8)³ 
as
(x-8)(x-8)(x-8) = 0
we would have
x-8=0; x-8=0; x-8=0
  x=8;   x=8;   x=8

and 8 would be a zero three times.

The last one won't factor, so we use the quadratic formula

4x%5E2-12x-1+=+0
x+=+%28-%28-12%29+%2B-+sqrt%28%28-12%29%5E2-4%284%29%28-1%29+%29%29%2F%282%284%29%29+
x+=+%2812+%2B-+sqrt%28144%2B16+%29%29%2F8+
x+=+%2812+%2B-+sqrt%28160%29%29%2F8+
x+=+%2812+%2B-+sqrt%2816%2A10%29%29%2F8+
x+=+%2812+%2B-+4sqrt%2810%29%29%2F8+
x+=+%284%283+%2B-+sqrt%2810%29%29%29%2F8+
x+=+%283+%2B-+sqrt%2810%29%29%2F2+

So the zeros are

8 with multiplicity 3
7i with multiplicity 1
-7i with multiplicity 1
%283+%2B+sqrt%2810%29%29%2F2+
with multiplicity 1
and
%283+-+sqrt%2810%29%29%2F2+
with multiplicity 1

Edwin