Question

write the factored form of the polynomial function with real coefficients, a lead coefficient of 1, and zeros of \\(-3, -2, 3\\), and \\(2\\).

\\y(x) = \\

Explanation:

⚡ Using what you learned: Graphing Polynomial Functions

Step 1: Identify the factors from the zeros

If \( r \) is a zero of a polynomial, then \( (x - r) \) is a factor of the polynomial.

Given zeros: \(-3, -2, 3, 2\)

• For zero \(-3\): the factor is \( (x - (-3)) = (x + 3) \)
• For zero \(-2\): the factor is \( (x - (-2)) = (x + 2) \)
• For zero \(3\): the factor is \( (x - 3) \)
• For zero \(2\): the factor is \( (x - 2) \)

Step 2: Write the factored form with the lead coefficient

The general factored form of a polynomial is:

$$ y(x) = a(x - r_1)(x - r_2)(x - r_3)(x - r_4) $$

Given that the lead coefficient \( a = 1 \):

$$ y(x) = 1(x + 3)(x + 2)(x - 3)(x - 2) $$

Simplifying the expression:

$$ y(x) = (x + 3)(x + 2)(x - 3)(x - 2) $$

Answer:

\( (x + 3)(x + 2)(x - 3)(x - 2) \)