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

Ask by Bates Davey. in the United States

Mar 31,2025

Question

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

Ask by Bates Davey. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

1. Since $$ 3-2i $$ is a zero of the polynomial
   $$
   P(x)=x^3-4x^2+x+26,
   $$
   the conjugate zeros theorem tells us that $$ 3+2i $$ is also a zero.

2. The factors corresponding to these zeros are
   $$
   (x-(3-2i))(x-(3+2i)).
   $$
   We compute their product:
   $$
   (x-(3-2i))(x-(3+2i)) = (x-3+2i)(x-3-2i).
   $$
   Recognize this as a product of conjugates, which simplifies to
   $$
   (x-3)^2-(2i)^2 = (x-3)^2+4.
   $$
   Thus, we have:
   $$
   (x-3)^2+4 = x^2-6x+9+4 = x^2-6x+13.
   $$

3. Since $$ P(x) $$ is of degree 3 and we already have a quadratic factor, the remaining factor must be linear. Assume
   $$
   P(x)=(x^2-6x+13)(x-\alpha).
   $$
   Expanding the product gives:
   $$
   (x^2-6x+13)(x-\alpha)=x^3-(\alpha+6)x^2+(6\alpha+13)x-13\alpha.
   $$
   Compare this with
   $$
   P(x)=x^3-4x^2+x+26.
   $$
   Matching the coefficients, we have:
   $$
   \alpha+6=4 \quad \Longrightarrow \quad \alpha=-2.
   $$

4. Therefore, the zeros of $$ P(x) $$ are:
   $$
   3-2i,\quad 3+2i,\quad -2.
   $$
The zeros of the polynomial $$ P(x) = x^3 - 4x^2 + x + 26 $$ are $$ 3-2i $$, $$ 3+2i $$, and $$ -2 $$.

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

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

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