To construct a polynomial [tex]\( P(x) \)[/tex] of degree 3 with the given properties, let's follow these steps:
1. Identify the roots and their multiplicities:
- There is a root of multiplicity 2 at [tex]\( x = 4 \)[/tex]. This means [tex]\((x - 4)^2\)[/tex] is a factor of the polynomial.
- There is a root of multiplicity 1 at [tex]\( x = -3 \)[/tex]. This means [tex]\((x + 3)\)[/tex] is a factor of the polynomial.
2. Form the polynomial with these factors:
Using the roots and their multiplicities, the polynomial can be initially written as:
[tex]\[
P(x) = C \cdot (x - 4)^2 \cdot (x + 3)
\][/tex]
where [tex]\( C \)[/tex] is a constant coefficient that we need to determine.
3. Determine the constant [tex]\( C \)[/tex] using the y-intercept:
The y-intercept is given as [tex]\((0, -14.4)\)[/tex], which means [tex]\( P(0) = -14.4 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the polynomial will help us find [tex]\( C \)[/tex]:
[tex]\[
P(0) = C \cdot (0 - 4)^2 \cdot (0 + 3)
\][/tex]
Simplify the expression:
[tex]\[
P(0) = C \cdot 16 \cdot 3 = 48C
\][/tex]
Given that [tex]\( P(0) = -14.4 \)[/tex], we can set up the equation:
[tex]\[
48C = -14.4
\][/tex]
4. Solve for [tex]\( C \)[/tex]:
[tex]\[
C = \frac{-14.4}{48} = -0.3
\][/tex]
5. Write the final polynomial:
Substituting [tex]\( C = -0.3 \)[/tex] back into the polynomial:
[tex]\[
P(x) = -0.3 \cdot (x - 4)^2 \cdot (x + 3)
\][/tex]
Thus, the polynomial [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = -0.3 (x - 4)^2 (x + 3)
\][/tex]
