Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function that has the given zeros. (There are many correct answers.) \(0,1,6\)

Step-by-Step Solution

Answer

A polynomial function with the roots \(0, 1, 6\) is \(f(x) = x^3 - 7x^2 + 12x\).

Step 1: Identify Polynomial Factors from Roots

Given roots are \(0, 1, 6\). The roots correlate with the factors of the polynomial in such a way that for every root \(r\), there is a factor \((x - r)\). So, the corresponding polynomial factors for these roots are \((x - 0)\), \((x - 1)\) and \((x - 6)\). Simplified factors are \(x\), \(x - 1\), and \(x - 6\).

Step 2: Generate the Polynomial Function

Create the polynomial function by multiplying these factors together. The polynomial function is then \(f(x) = (x)(x - 1)(x - 6)\).

Step 3: Expand the Polynomial Function

Expand these factors to have the polynomial in standard form. Expand \(f(x) = (x)(x - 1)(x - 6)\) to \(f(x) = x^3 - 7x^2 + 12x\). This is the resultant polynomial.

Key Concepts

Roots of Polynomials

When dealing with polynomial functions, the roots (or zeros) are crucial because they determine where the polynomial equals zero. In simpler terms, a root of a polynomial is a value for which the polynomial becomes zero. For example, if you plug the root into the polynomial equation, the result should be zero.

Identifying roots is an essential part of understanding polynomials. They can provide insights into the behavior of the graph of the polynomial function. Roots can be real or complex:

• Real roots are those that lie on the real number line and can be found within the polynomial's equation directly.
• Complex roots, on the other hand, involve imaginary numbers and occur in conjugate pairs.

In our specific example with roots 0, 1, and 6, the understanding of roots helps in forming the corresponding factors and thus the base of our polynomial function.

Factorization

Factorization is the process of breaking down a polynomial into a product of its factors. Each factor corresponds to a root of the polynomial. This is often the first step when you're given roots and need to find a polynomial function.

For each root given, we can create a linear factor of the form \( (x - r) \), where \( r \) is the root. So, for our roots 0, 1, and 6, the factors are \( x \), \( (x - 1) \), and \( (x - 6) \), respectively.

By multiplying these factors, we get the initial polynomial expression. Factorization simplifies complex expressions and is a fundamental skill in algebra.

• Identifying linear factors from roots is straightforward.
• Multiplying the factors gives the developing form of the polynomial.
• Simplification "cleans up" the polynomial into standard form.

Therefore, factorization is an effective way to understand the structure and start building the polynomial from its roots.

Standard Form of a Polynomial

Once you have factorized the polynomial using its roots, the next step is to express it in the standard form. A polynomial in standard form is expressed as \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are coefficients and \( a_n eq 0 \). This form organizes the polynomial by descending powers of \( x \).

To achieve this from the factored form, expand the products of the factors. For example, given \( f(x) = (x)(x - 1)(x - 6) \):

• Start by multiplying any two factors:
  • \( (x)(x - 1) = x^2 - x \)
• Then multiply by the remaining factor:
  • \( (x^2 - x)(x - 6) = x^3 - 7x^2 + 12x \)

The polynomial is now expanded into its standard form. This visual representation of the polynomial helps in understanding its degree and shape, and it is essential for graphing the function.