Problem 34 - Algebra 2

Question

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(-2,2,4,6\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = x^4 - 10x^3 + 20x^2 + 40x - 96 \).

Step 1: Use Zeros to Form Factors

First, recall that if a polynomial has a zero at a number \(a\), then \((x-a)\) is a factor of that polynomial. Thus, using the given zeros \(-2, 2, 4, \) and \(6\), we can form the factors: \((x + 2), (x - 2), (x - 4),\) and \((x - 6)\).

Step 2: Write the Polynomial Function

Combine the factors from Step 1 to write a polynomial function: \[ f(x) = (x + 2)(x - 2)(x - 4)(x - 6) \]. Since all roots are real and distinct, this is the polynomial of least degree.

Step 3: Expand the Polynomial

Expand the polynomial to express it in standard form. Start by expanding pairs of factors: \( (x + 2)(x - 2) = x^2 - 4 \) (difference of squares), and \( (x - 4)(x - 6) = x^2 - 10x + 24 \).

Step 4: Multiply the Results from Step 3

Now multiply the results from Step 3: \[ (x^2 - 4)(x^2 - 10x + 24) = x^4 - 10x^3 + 24x^2 - 4x^2 + 40x - 96 = x^4 - 10x^3 + 20x^2 + 40x - 96 \].

Step 5: Simplify the Expression

Combine like terms to simplify the expression if needed. The polynomial is already in its simplest form: \( f(x) = x^4 - 10x^3 + 20x^2 + 40x - 96 \).

Key Concepts

Integral Coefficients

In the world of polynomials, integral coefficients refer to the numbers that multiply the variables and are integers. These are simple whole numbers without fractions or decimals.
They play a crucial role when writing polynomial functions, especially when asked to have a polynomial with integral coefficients. This constraint ensures that all the component numbers making up the polynomial remain as integers, which often makes the polynomial easier to work with.

• For example, in the polynomial function derived from given zeros: \[ f(x) = (x + 2)(x - 2)(x - 4)(x - 6) \]each coefficient is indeed an integer after expansion.
• Such polynomials are very suitable in practical applications since integer arithmetic is more straightforward than dealing with fractions or decimals.

Remember, ensuring integral coefficients means that during operations such as multiplication or addition, you carefully maintain whole numbers throughout the process by properly combining the terms.

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In simpler terms, they are the roots of the polynomial, the points where the graph of the polynomial crosses the x-axis.
In the given exercise, the zeros are

• -2
• 2
• 4
• 6

These are the numbers where the polynomial \(f(x)\) becomes zero. Finding zeros is a pivotal process because they assist in forming the polynomial itself.
To turn these zeros into factors, we use the concept that if \(a\) is a zero, then \((x - a)\) is a factor. Therefore, for the zeros

• -2 resulting in the factor \((x + 2)\)
• 2 resulting in the factor \((x - 2)\)
• 4 resulting in the factor \((x - 4)\)
• 6 resulting in the factor \((x - 6)\)

By multiplying these factors, we start building the polynomial that aligns with these zeros.

Factoring Polynomials

Factoring polynomials is the reverse of expanding, where we rewrite the polynomial as a product of its factors. It boils down a large polynomial into simpler, smaller building blocks.
In our context, we reached the polynomial: \[ f(x) = (x + 2)(x - 2)(x - 4)(x - 6) \]by identifying each factor that matched the given zeros. Here, factoring used the relationship between zeros and factors which enabled us to express the polynomial as the product of four linear factors.

• This strategy is essential when solving equations or simplifying expressions, as it helps reveal important properties such as zeros or intercepts.
• When polynomials are presented with integral coefficients, it ensures the factors can be reconstructed into simple, understandable parts.
• It's especially convenient for solving polynomial equations, finding roots, or simplifying complex expressions.
• Not every polynomial can be easily factored, but in situations where it's possible, it significantly streamlines the problem-solving process.

Understanding factoring is key since it crosses paths with various operations in algebra, helping to solve equations efficiently and simplifying one's workflow.