Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

Verified step by step guidance

1. Step 1: Identify the x-intercepts (roots) of the polynomial from the graph. The graph crosses the x-axis at approximately x = -5, x = 1, and x = 4. These are the zeros of the polynomial.
2. Step 2: Determine the multiplicity of each root by observing the behavior of the graph at each x-intercept. Since the graph crosses the x-axis at each root (not just touches), each root has an odd multiplicity, most likely 1 for the least degree polynomial.
3. Step 3: Write the general form of the polynomial using the roots. Since the roots are -5, 1, and 4, the polynomial can be expressed as \(f(x) = a(x + 5)(x - 1)(x - 4)\), where \(a\) is a constant coefficient to be determined.
4. Step 4: Use the given point on the graph, which is the y-intercept at (0, 20), to find the value of \(a\). Substitute \(x = 0\) and \(f(0) = 20\) into the polynomial: \$20 = a(0 + 5)(0 - 1)(0 - 4)$.
5. Step 5: Solve the equation from Step 4 for \(a\) to find the leading coefficient. Then write the final polynomial function \(f(x)\) by substituting \(a\) back into the general form.

Key Concepts

Polynomial Function and Degree

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. The degree of the polynomial is the highest exponent of the variable, which determines the general shape and number of turning points of the graph.

Zeros and Multiplicity

Zeros of a polynomial are the x-values where the function equals zero, corresponding to x-intercepts on the graph. The multiplicity of a zero affects the graph's behavior at that point: odd multiplicities cross the x-axis, while even multiplicities touch and turn around without crossing.

Using Points to Determine Coefficients

Known points on the graph, such as the y-intercept, help determine the coefficients of the polynomial. Substituting these points into the polynomial equation allows solving for unknown constants, ensuring the polynomial fits the given graph accurately.