Question

the polynomial of degree 3, (p(x)), has a root of multiplicity 2 at (x = 3) and a root of multiplicity 1 at (x = -2). the (y)-intercept is (y = -5.4).
find a formula for (p(x)).
(p(x) =)

question 8
given the function (p(x) = (x - 3)^2 (x - 5)), find
its (y)-intercept is
its (x)-intercepts are (x_1 =) and (x_2 =) with (x_1 < x_2)
when (x \to infty), (y \to infty) (input + or - for the answer)
when (x \to -infty), (y \to infty) (input + or - for the answer)

Explanation:

🆕 New Concept Discovered: Writing Polynomials from Roots and Multiplicities
Building equations using roots, exponents, and a given point.

Step 1: Set up the general form of the polynomial

A root at \( x = r \) with multiplicity \( k \) corresponds to a factor of the form \( (x - r)^k \) in the polynomial.

We are given:

• A root of multiplicity 2 at \( x = 3 \), which gives the factor \( (x - 3)^2 \).
• A root of multiplicity 1 at \( x = -2 \), which gives the factor \( (x - (-2))^1 = (x + 2) \).

We include a leading coefficient \( a \) to represent any vertical stretch or compression:

$$ P(x) = a(x - 3)^2(x + 2) $$

Step 2: Solve for the leading coefficient \( a \)

We are given that the \( y \)-intercept is \( y = -5.4 \). This means that when \( x = 0 \), \( P(0) = -5.4 \).

Substitute \( x = 0 \) and \( P(0) = -5.4 \) into our equation:

$$ -5.4 = a(0 - 3)^2(0 + 2) $$

$$ -5.4 = a(-3)^2(2) $$

$$ -5.4 = a(9)(2) $$

$$ -5.4 = 18a $$

Now, solve for \( a \):

$$ a = \frac{-5.4}{18} $$

$$ a = -0.3 $$

Step 3: Write the final formula

Substitute \( a = -0.3 \) back into the polynomial equation:

$$ P(x) = -0.3(x - 3)^2(x + 2) $$

Answer:

$$ P(x) = -0.3(x - 3)^2(x + 2) $$