Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. See Example 4. Zero of -3 having multiplicity 3; ƒ(3)=36

Verified step by step guidance

1. Start by understanding that a polynomial of degree 3 with a zero at x = -3 and multiplicity 3 can be expressed as f(x) = a(x + 3)^3.
2. Use the given condition f(3) = 36 to find the value of the coefficient 'a'.
3. Substitute x = 3 into the polynomial: f(3) = a(3 + 3)^3.
4. Simplify the expression: f(3) = a(6)^3.
5. Set the expression equal to 36 and solve for 'a': a(6)^3 = 36.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of a polynomial is determined by the highest power of the variable. In this case, a degree 3 polynomial will have the general form ƒ(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are real coefficients.

Multiplicity of Roots

Multiplicity refers to the number of times a particular root appears in a polynomial. A root with multiplicity 3 means that the polynomial can be expressed as (x + 3)³, indicating that -3 is a root that contributes three times to the polynomial's behavior. This affects the shape of the graph, causing it to touch the x-axis at -3 without crossing it.

Evaluating Polynomial Functions

Evaluating a polynomial function involves substituting a specific value for the variable and calculating the result. In this problem, we need to ensure that the polynomial satisfies the condition ƒ(3) = 36, meaning when we substitute x = 3 into our polynomial, the output must equal 36. This condition helps in determining the coefficients of the polynomial.