Question

The function f(x) = -x5 +78x3 - 64x2 - 357x - 216 has the

graph given to the right. Use the graph to factor the polynomial.

Answer 1

`f(x) = -(x+9)(x-3)(x-8)(x+1)^2`

Explanation:

The graph intersects the x-axis at
`x=-9,3,8`, so there are factors of
`(x+9)`,
`(x-3)`, and
`(x-8)`

The graph touches the x-axis at
`x=-1`, which indicates that there is a multiplicity. This multiplicity is 2 because there must be 5 roots and 3 were found already, making the remaining factor
`(x+1)^2`.

As the original polynomial has a negative leading coefficient, so should our factored polynomial. Therefore, the factored polynomial is
`-(x+9)(x-3)(x-8)(x+1)^2`.