The function $-x^{5}+36 x^{3}-22 x^{2}-147 x-90$ has the graph given to the right. Use the graph to factor the polynomial.
What is the factored form of the polynomial?
Answer
So, the factored form of the polynomial is \(\boxed{-x(x+3)(x+2)(x-1)(x-3)(x-5)}\).
Step 1 ：First, we need to find the roots of the polynomial from the graph. The roots are the x-values where the function crosses the x-axis.
Step 2 ：From the graph, we can see that the roots are -3, -2, 1, 3, and 5.
Step 3 ：So, we can write the polynomial as \(-x^{5}+36 x^{3}-22 x^{2}-147 x-90 = -x(x+3)(x+2)(x-1)(x-3)(x-5)\).
Step 4 ：We can check this by expanding the factored form and simplifying to see if we get the original polynomial.
Step 5 ：After checking, we find that the factored form is correct.
Step 6 ：So, the factored form of the polynomial is \(\boxed{-x(x+3)(x+2)(x-1)(x-3)(x-5)}\).
