Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=−x³+x²+16x−16

Verified step by step guidance

1. Identify the polynomial function given: $f\left(x\right)=-x^3+x^2+16x-16$.
2. To find the zeros of the function, set $f\left(x\right)=\; 0$ and solve the cubic equation $-x^3\; +\; x^2\; +\; 16x\; -\; 16\; =\; 0$.
3. Look for rational roots using the Rational Root Theorem, which suggests testing factors of the constant term (±1, ±2, ±4, ±8, ±16) as possible zeros.
4. Use synthetic division or polynomial division to test each candidate root. When a root is found, factor it out to reduce the cubic to a quadratic.
5. Solve the resulting quadratic equation using factoring, completing the square, or the quadratic formula to find the remaining zeros. These zeros will help in sketching the complete graph.

Key Concepts

Finding Zeros of Polynomial Functions

Zeros of a polynomial are the values of x for which the function equals zero. To find them, set f(x) = 0 and solve the resulting equation using factoring, synthetic division, or the Rational Root Theorem. These zeros correspond to the x-intercepts of the graph.

Polynomial Function Behavior and End Behavior

The degree and leading coefficient of a polynomial determine its end behavior, or how the graph behaves as x approaches positive or negative infinity. For example, a cubic with a negative leading coefficient falls to the right and rises to the left, guiding the overall shape of the graph.

Sketching Polynomial Graphs Without Technology

To sketch a polynomial graph by hand, identify zeros, determine their multiplicities, analyze end behavior, and find key points such as local maxima and minima using derivatives or sign changes. This approach helps create an accurate, complete graph without graphing utilities.