Problem 57 - Algebra and Trigonometry

Question

Find a polynomial of the specified degree that has the given zeros. Degree \(3 ; \quad\) zeros \(-1,1,3\)

Step-by-Step Solution

Answer

The polynomial is \(f(x) = x^3 - 3x^2 - x + 3\).

Step 1: Identify the Form of the Polynomial

Since the polynomial is of degree 3 with zeros -1, 1, and 3, we know we can express the polynomial as the product of three linear factors corresponding to these zeros. Given the zeros are \(-1\), \(1\), and \(3\), the polynomial can be written in the form: \[ f(x) = a(x + 1)(x - 1)(x - 3) \]where \(a\) is a non-zero constant.

Step 2: Expand the Factors

We need to expand the product of the linear factors to write the polynomial in standard form. The expression \[(x + 1)(x - 1)(x - 3)\]can first be expanded by using the identity \((x + 1)(x - 1) = x^2 - 1\).Including the third factor, we expand \[(x^2 - 1)(x - 3)\].

Step 3: Multiply to Expand Further

Continuing from the previous step, we perform the multiplication: \[(x^2 - 1)(x - 3) = x^2(x - 3) - 1(x - 3)\].This results in: \[x^3 - 3x^2 - x + 3\].

Step 4: Assign the Leading Coefficient

Since the problem does not specify a leading coefficient other than that for a general polynomial, we can take \(a = 1\). Thus, the polynomial is:\[ f(x) = x^3 - 3x^2 - x + 3 \].

Key Concepts

degree of a polynomial

The degree of a polynomial is an important concept to understand, as it defines the highest power of the variable in the polynomial. In simpler words, it's the largest exponent present in the polynomial expression. For example, consider the polynomial \( f(x) = x^3 - 3x^2 - x + 3 \). Here, the term \( x^3 \) has the highest power of 3, thus the degree of this polynomial is 3.

• The degree can tell us a lot about the polynomial's behavior.
• A polynomial of degree 3 will have at most three zeros.
• The graph of a third-degree polynomial can have up to two turning points.

Understanding the degree is crucial because it influences the polynomial's shapes and how it stretches or shrinks when plotted on a graph.

zeros of a polynomial

Zeros of a polynomial, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. If you have a polynomial \( f(x) \), the zeros are the values for which \( f(x) = 0 \). In the exercise given, the polynomial \( f(x) = x^3 - 3x^2 - x + 3 \) has zeros at \(-1, 1,\) and \(3\).

• Zeros are essential because they help in determining where the graph of the polynomial will intersect the x-axis.
• Finding zeros is often the key step in graphing a polynomial.
• If you know the zeros, you can construct the polynomial by using these as the basis for the linear factors.

Identifying zeros allows us to understand the structure and characteristics of the polynomial.

linear factors

Once you've identified the zeros of a polynomial, you can use them to write the polynomial as a product of linear factors. Each zero \( r \) corresponds to a linear factor of the form \( (x - r) \). For the zeros \(-1, 1,\) and \(3\), the corresponding linear factors are \( (x + 1), (x - 1),\) and \((x - 3)\). Thus, the polynomial can be expressed as:
\[ f(x) = a(x + 1)(x - 1)(x - 3) \]

• The constant \( a \) can be any non-zero number, often set to 1 for simplicity unless specified otherwise.
• This representation makes it easier to factorize the polynomial and find all the zeros.
• By multiplying these linear factors, you can achieve the standard form of the polynomial.

Linear factors are foundational in polynomial algebra because they directly relate to the polynomial's zeros.

expansion of polynomials

Expanding a polynomial from its linear factors means you're transforming it from its factored form back into its standard form. This process involves multiplying the linear factors together. For \( f(x) = (x + 1)(x - 1)(x - 3) \), you expand this by first simplifying \( (x + 1)(x - 1) = x^2 - 1 \), then continue by multiplying \( (x^2 - 1)(x - 3) \) to arrive at \( f(x) = x^3 - 3x^2 - x + 3 \).

• The expansion involves several steps of multiplication, often using distributive properties.
• This standard form of a polynomial is easier to differentiate and integrate.
• It also allows you to easily identify the leading term and the degree of the polynomial.

Expanding polynomials aids in simplifying expressions for further algebraic manipulation.