3. Write a polynomial in factored form with zeros $$ -1,-2,-3 $$ (double root) and a $$ y $$-intercept of $$ (0,-18) $$.

Question

UpStudy AI · Accepted Answer

To construct a polynomial with the given conditions, follow these steps:

1. **Identify the Zeros and Their Multiplicities:**
   - **Zeros:** $$ -1 $$, $$ -2 $$, $$ -3 $$
   - **Multiplicity:** The zero $$ -3 $$ is a double root, meaning it appears twice.

2. **Write the Polynomial in Factored Form:**
   - Each zero corresponds to a factor of the polynomial.
   - Since $$ -3 $$ is a double root, its factor is squared.
   $$
   P(x) = a(x + 1)(x + 2)(x + 3)^2
   $$
   Here, $$ a $$ is a constant that we'll determine using the $$ y $$-intercept.

3. **Determine the Constant $$ a $$ Using the $$ y $$-Intercept:**
   - The $$ y $$-intercept is given as $$ (0, -18) $$, which means $$ P(0) = -18 $$.
   - Substitute $$ x = 0 $$ into the polynomial:
     $$
     -18 = a(0 + 1)(0 + 2)(0 + 3)^2
     $$
     $$
     -18 = a(1)(2)(9)
     $$
     $$
     -18 = 18a
     $$
     $$
     a = -1
     $$

4. **Write the Final Polynomial:**
   $$
   P(x) = -1(x + 1)(x + 2)(x + 3)^2
   $$
   This simplifies to:
   $$
   P(x) = -(x + 1)(x + 2)(x + 3)^2
   $$

**Final Answer:**
$$
y = -(x + 1)(x + 2)(x + 3)^2
$$
The polynomial in factored form is $$ y = -(x + 1)(x + 2)(x + 3)^2 $$.

3. Write a polynomial in factored form with zeros \( -1,-2,-3 \) (double root) and a \( y \)-intercept of \( (0,-18) \).

Real Tutor Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions