Question

Write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros. \(-4,-2,5\)

Short answer

The polynomial function of least degree with leading coefficient 1 and having the zeros -4, -2, and 5 is \( f(x) = x^3 + x^2 -22x -40 \).

Step-by-step solution

Understand the concept of zeros

Zeros are the x-values where the function equals to zero. These values make the equation equal to zero. In this exercise, the zeros given are -4, -2, and 5.

Write the polynomial in factored form

A polynomial function can be written in factored form as \( f(x) = a(x-r1)(x-r2)...(x-rn) \), where \( a \) is the leading coefficient, and \( r1, r2, ..., rn \) are the zeros. Since the leading coefficient is 1 in this exercise, the polynomial function with zeros -4, -2 and 5 can be written as \( f(x) = (x-(-4))(x-(-2))(x-5) \).

Simplification

Simplify the expression to get the polynomial function in standard form. \( f(x) = (x+4)(x+2)(x-5) \)

Multiplication of the Factors

Multiply out the factors to obtain the polynomial function. This results in \( f(x) = x^3 + x^2 -22x -40 \).

Key concepts

Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest degree. It plays an important role in determining the shape and direction of the polynomial's graph.
For instance, in the polynomial function, if the leading coefficient is positive, the ends of the graph of the polynomial will point upwards. Conversely, if it is negative, the ends will point downwards.
In this exercise, we're asked to find a polynomial with a leading coefficient of 1, which simplifies the process of constructing the polynomial since it eliminates the need for additional scaling.

• This means the function is essentially dictated by the zero values provided, making it easier to directly identify the polynomial in factored form.

Zeros of a Polynomial

Zeros of a polynomial, also known as roots or solutions, are values of x that satisfy the equation when the function equals zero. In simple terms, they are the x-values where the graph of the polynomial touches or crosses the x-axis.
To identify a polynomial's zeros, you factor the polynomial and set each factor equal to zero.
In the context of this exercise, the zeros are given as -4, -2, and 5, which means these values will be substituted into the factored form of the polynomial as \( f(x) = (x+4)(x+2)(x-5) \).

Factored Form

The factored form of a polynomial is an expression showing the polynomial as a product of its factors. This form is particularly useful for finding the zeros of the polynomial.
For example, a polynomial with zeros \( r1, r2, \) and \( r3 \) can be expressed as \( f(x) = (x-r1)(x-r2)(x-r3) \). In our exercise, with a leading coefficient of 1, the factored form becomes \( f(x) = (x+4)(x+2)(x-5) \).
This form makes it easier to visualize and manipulate the roots, offering a straightforward method to find the solutions when set to zero.

• The main advantage of the factored form is its simplicity in representing complex polynomials in terms of their basic building blocks, i.e., linear factors.

Standard Form

The standard form of a polynomial is when it is expressed as a sum of terms ordered by descending powers of the variable.
Each term consists of a coefficient and a variable raised to a degree. In our problem, after multiplying out the factored form \( (x+4)(x+2)(x-5) \), the resulting polynomial is represented in its standard form: \( f(x) = x^3 + x^2 - 22x - 40 \).
This standard form is straightforward for evaluating the polynomial at any given x-value and is often preferred for analytical or computational purposes.

• It provides a clear overview of the polynomial's degree and the influence of each term on the graph of the polynomial.
• By working in standard form, it is easier to apply calculus techniques such as differentiation or integration for further analysis.