Question

Please help me with this

Answer 1

The equation
`f(x) = - (1)/(32)\cdot (x + 4) \cdot (x + 2) \cdot (x - 1) \cdot (x + 4)^2` represents the polynomial graphed.

How to determine the equation of a polynomial

In this problem we must determine the equation of a polynomial, whose factor form is now defined:

f(x) = a · Π (x - rₙ), for {1, 2, 3, ..., n - 1, n}

Where a is the lead coefficient and rₙ is the n-th root.

If we know that r₁ = - 4, r₂ = - 2, r₃ = 1, r₄ = 4 (Multiplicity 2), Intercept: 4, then the polynomial is:

f(x) = a · (x + 4) · (x + 2) · (x - 1) · (x + 4)²

And the lead coefficient is:

4 = a · (0 + 4) · (0 + 2) · (0 - 1) · (0 + 4)²

4 = a · 4 · 2 · (- 1) · 16

4 = - 128 · a

`a = - (1)/(32)`

The polynomial is defined by
`f(x) = - (1)/(32)\cdot (x + 4) \cdot (x + 2) \cdot (x - 1) \cdot (x + 4)^2`.