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)=4x³−8x²−3x+9

Verified step by step guidance

1. Start by finding the zeros of the polynomial function $f\left(x\right)=4x^3-8x^2-3x+9$. To do this, set $f\left(x\right)=\; 0$ and solve for $x$: $4x^3\; -\; 8x^2\; -\; 3x\; +\; 9\; =\; 0$.
2. Use the Rational Root Theorem to list possible rational zeros. The possible rational roots are factors of the constant term (9) divided by factors of the leading coefficient (4). So, possible roots include ±1, ±3, ±9, ±1/2, ±3/2, ±9/2, ±1/4, ±3/4, ±9/4.
3. Test these possible roots by substituting them into the polynomial or by using synthetic division to check if they yield zero. When you find a root, say $r$, factor out $(x\; -\; r)$ from the polynomial.
4. After factoring out one root, you will get a quadratic polynomial. Solve this quadratic either by factoring, completing the square, or using the quadratic formula to find the remaining zeros.
5. Once all zeros are found, analyze the multiplicity of each zero to understand the behavior of the graph at those points. Then, use this information along with the end behavior of the cubic function (considering the leading coefficient and degree) to sketch a complete graph without a graphing utility.

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 the polynomial equal to zero and solve 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

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

Sketching Polynomial Graphs Without Technology

To sketch a polynomial graph, identify zeros, determine their multiplicities, find critical points using derivatives or sign analysis, and use end behavior to complete the graph. This approach helps visualize the function’s shape without graphing tools.