Using the conjugate zeros theorem to find all zeros of a polynomial
Find all other zeros of , given that is a zero.
(If there is more than one zero, separate them with commas.)
(

Ask by O'Quinn Bright. in the United States

Mar 31,2025

Question

Using the conjugate zeros theorem to find all zeros of a polynomial
Find all other zeros of , given that is a zero.
(If there is more than one zero, separate them with commas.)
(

Ask by O'Quinn Bright. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

1. Since the polynomial
   $$
   P(x)=x^{3}-2x^{2}-6x-8
   $$
   has real coefficients and one complex zero $$ -1-i $$, by the conjugate zeros theorem, its conjugate $$ -1+i $$ is also a zero.

2. The zeros $$ -1-i $$ and $$ -1+i $$ correspond to the quadratic factor obtained by:
   $$
   (x-(-1-i))(x-(-1+i)) = (x+1+i)(x+1-i).
   $$
   Multiplying these factors:
   $$
   (x+1+i)(x+1-i) = (x+1)^2 - i^2 = (x+1)^2 + 1.
   $$
   Expanding, we get:
   $$
   (x+1)^2 + 1 = x^2+2x+1+1 = x^2+2x+2.
   $$

3. Since $$ x^2+2x+2 $$ is a factor of $$ P(x) $$, we can write:
   $$
   P(x)= (x^2+2x+2)(Q(x)),
   $$
   where $$ Q(x) $$ is the remaining linear factor. Let
   $$
   Q(x) = ax+b.
   $$
   Multiplying out:
   $$
   (x^2+2x+2)(ax+b) = ax^3 + (2a+b)x^2 + (2a+2b)x+2b.
   $$
   Equate with $$ P(x) = x^3-2x^2-6x-8 $$ by matching coefficients:

- Coefficient of $$ x^3 $$: $$ a=1 $$.
   - Coefficient of $$ x^2 $$: $$ 2a+b = -2 $$ implies $$ 2(1)+b=-2 $$ so $$ b=-4 $$.
   - Verifying the other coefficients:
     - Coefficient of $$ x $$: $$ 2a+2b = 2(1)+2(-4)= 2-8=-6 $$,
     - Constant term: $$ 2b=2(-4)=-8 $$.

4. Hence, the factorization of $$ P(x) $$ is:
   $$
   P(x)= (x^2+2x+2)(x-4).
   $$
   The remaining zero comes from $$ x-4=0 $$, giving:
   $$
   x=4.
   $$

The zeros of the polynomial $$ P(x) $$ are $$ -1-i $$, $$ -1+i $$, and $$ 4 $$.
The zeros of the polynomial $$ P(x) = x^{3} - 2x^{2} - 6x - 8 $$ are $$ -1 - i $$, $$ -1 + i $$, and $$ 4 $$.

Using the conjugate zeros theorem to find all zeros of a polynomial
Find all other zeros of , given that is a zero.
(If there is more than one zero, separate them with commas.)
(

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions

Using the conjugate zeros theorem to find all zeros of a polynomial Find all other zeros of , given that is a zero. (If there is more than one zero, separate them with commas.) (

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions

Using the conjugate zeros theorem to find all zeros of a polynomial
Find all other zeros of , given that is a zero.
(If there is more than one zero, separate them with commas.)
(