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

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)}}}

Question 1156238: Identify the degree of the polynomial. Find all zeros (real and imaginary) and indicate their multiplicity if greater than 1.

Answer by Edwin McCravy(20060) (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:



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:

 



(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










So the zeros are

8 with multiplicity 3
7i with multiplicity 1
-7i with multiplicity 1

with multiplicity 1
and

with multiplicity 1

Edwin

RELATED QUESTIONS
State the degree of the following polynomial equation. Find all of the real and imaginary (answered by ewatrrr)

find all real zeros and their multiplicity of the polynomial function... (answered by Fombitz)

If x=-1 and x=2 are two zeros of the polynomial given by P(x)=x^4-x^3-3x^2+x+2, find... (answered by richwmiller)

Find all real and imaginary zeros of the following polynomial equation: m(x)=... (answered by stanbon)

Find a polynomial f(x) of degree 4 with real coefficients and the following zeros... (answered by ikleyn)

Find a polynomial f(x) of degree 4 with real coefficients and the following zeros. 1,... (answered by MathLover1)

Form a polynomial f(x) with real coefficients having the given degree and zeros Degree (answered by ikleyn)

Factor the polynomial completely. {{{Q(x) = x^2 + 4x + 5}}} Find all its zeros. State (answered by josgarithmetic)

Factor the polynomial completely. P(x) = x3 + 25x Find all its zeros. State the... (answered by Fombitz)