Question 1156238
<pre>
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^7-108x^6+1251x^5-9620x^4+57647x^3-211560x^2+291648x + 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(x)=(x  -  8)^3(x^2 +49)(4x^2  -  12x  -  1)}}} 

{{{(x^""  -  8)^3(x^2 +49)(4x^2  -  12x  -  1)=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^2-12x-1 = 0}}}
{{{x = (-(-12) +- sqrt((-12)^2-4(4)(-1) ))/(2(4)) }}}
{{{x = (12 +- sqrt(144+16 ))/8 }}}
{{{x = (12 +- sqrt(160))/8 }}}
{{{x = (12 +- sqrt(16*10))/8 }}}
{{{x = (12 +- 4sqrt(10))/8 }}}
{{{x = (4(3 +- sqrt(10)))/8 }}}
{{{x = (3 +- sqrt(10))/2 }}}

So the zeros are

8 with multiplicity 3
7i with multiplicity 1
-7i with multiplicity 1
{{{(3 + sqrt(10))/2 }}}
with multiplicity 1
and
{{{(3 - sqrt(10))/2 }}}
with multiplicity 1

Edwin</pre>