Question

the polynomial of degree 3, p(x), has a root of multiplicity 2 at x = 5 and a root of multiplicity 1 at x = -3. the y-intercept is y = -7.5. find a formula for p(x). p(x) =

Explanation:

Step1: Write the general form of the polynomial

Since \( P(x) \) is a degree 3 polynomial with a root of multiplicity 2 at \( x = 5 \) and a root of multiplicity 1 at \( x=-3 \), we can write it in factored form as \( P(x)=a(x - 5)^2(x + 3) \), where \( a \) is a constant coefficient.

Step2: Use the y - intercept to find the value of \( a \)

The y - intercept is the value of \( P(x) \) when \( x = 0 \). We know that the y - intercept is \( y=-7.5=-\frac{15}{2} \).
Substitute \( x = 0 \) into the equation \( P(x)=a(x - 5)^2(x + 3) \):
\[

$$\begin{align*} P(0)&=a(0 - 5)^2(0 + 3)\\ -7.5&=a\times(- 5)^2\times3\\ -7.5&=a\times25\times3\\ -7.5&=75a \end{align*}$$

\]
To solve for \( a \), divide both sides of the equation by 75:
\[
a=\frac{-7.5}{75}=-\frac{7.5}{75}=-\frac{1}{10}=-0.1
\]

Step3: Write the final formula for \( P(x) \)

Substitute \( a =-\frac{1}{10} \) into the factored form \( P(x)=a(x - 5)^2(x + 3) \):
\[
P(x)=-\frac{1}{10}(x - 5)^2(x + 3)
\]
We can also expand it:
First, expand \( (x - 5)^2=x^{2}-10x + 25 \)
Then, \( (x^{2}-10x + 25)(x + 3)=x^{3}+3x^{2}-10x^{2}-30x + 25x + 75=x^{3}-7x^{2}-5x + 75 \)
Multiply by \( -\frac{1}{10} \):
\[
P(x)=-\frac{1}{10}x^{3}+\frac{7}{10}x^{2}+\frac{1}{2}x-\frac{15}{2}
\]
Or we can leave it in factored form as \( P(x)=-\frac{1}{10}(x - 5)^2(x + 3) \)

Answer:

\( P(x)=-\frac{1}{10}(x - 5)^2(x + 3) \) (or the expanded form \( P(x)=-\frac{1}{10}x^{3}+\frac{7}{10}x^{2}+\frac{1}{2}x-\frac{15}{2} \))