Write the polynomial as a
product of linear factors.

Ask by Wheeler Estrada. in the United States

Jan 21,2025

Question

Write the polynomial as a product of linear factors. $$ x^{3}-x^{2}+9 x-9 $$ $$ (x-[?])(x-3 i)(x+[\quad i) $$

UpStudy AI · Accepted Answer

To express the polynomial $$ x^{3} - x^{2} + 9x - 9 $$ as a product of linear factors, follow these steps:

1. **Find the Real Root:**

Use the Rational Root Theorem to test possible rational roots. Possible candidates are $$ \pm1, \pm3, \pm9 $$.

- **Test $$ x = 1 $$:**
     $$
     1^{3} - 1^{2} + 9(1) - 9 = 1 - 1 + 9 - 9 = 0
     $$
     So, $$ x = 1 $$ is a root.

2. **Factor Out the Real Root:**

Divide the polynomial by $$ (x - 1) $$ to find the remaining quadratic factor.

$$
   \begin{array}{r|rrrr}
     1 & 1 & -1 & 9 & -9 \
       &   & 1  & 0 & 9  \
     \hline
       & 1 &  0 & 9 & 0
   \end{array}
   $$
   
   The quotient is $$ x^{2} + 9 $$.

3. **Factor the Quadratic:**

The quadratic $$ x^{2} + 9 $$ can be factored using imaginary numbers:
   $$
   x^{2} + 9 = (x - 3i)(x + 3i)
   $$
   
4. **Combine All Factors:**

Putting it all together, the polynomial can be expressed as:
   $$
   x^{3} - x^{2} + 9x - 9 = (x - 1)(x - 3i)(x + 3i)
   $$

**Final Answer:**
$$
x^{3} - x^{2} + 9x - 9 = (x - 1)(x - 3i)(x + 3i)
$$
$$ x^{3} - x^{2} + 9x - 9 = (x - 1)(x - 3i)(x + 3i) $$

Write the polynomial as a
product of linear factors.

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