Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{3}+10 x^{2}+33 x+34$$

Step-by-Step Solution

Answer

The zeros of the function are \(x = -1, -2, -17\). In factorized form, the polynomial is \((x + 1)(x + 2)(x + 17)\). These zeros correspond to the x-intercepts of the function's graph, which can be confirmed using a graphing tool.

Step 1: Finding the Zeros of the Function

In order to find the zeros of the function, set \(f(x) = 0\) and solve for \(x\). This gives us the following equation to solve: \(x^{3}+10 x^{2}+33 x+34 =0\). By using either factorization or synthetic division, we will be able to find the zeros of the function.

Step 2: Factorizing the Polynomial

When we factorize the provided polynomial, we get \((x + 1)(x + 2)(x + 17)=0\). These are the linear factors of the polynomial.

Step 3: Determining the x-intercepts from the Factors

The x-intercepts of the function can be determined from the factors. Setting each factor equal to zero and solving for \(x\) gives \(x = -1, -2, -17\). These are the three x-intercepts of the function.

Step 4: Verifying the x-intercepts using a Graphing Tool

To verify that these are the real zeros and the only x-intercepts, we can use a graphing tool to graph the function \(f(x)\) and observe where it intersects with the x-axis. The graphing tool should show that the function intersects the x-axis at the points we identified.

Key Concepts

Finding Zeros

Finding the zeros of a polynomial function is one of the fundamental tasks in algebra. Zeros are the values of the variable, usually denoted as \(x\), for which the polynomial evaluates to zero. For a given polynomial \(f(x) = x^3 + 10x^2 + 33x + 34\), we find its zeros by setting \(f(x) = 0\) and solving the equation.

• This involves looking for values of \(x\) that make the expression equal zero.
• Zeros are critical as they provide points where the graph of the polynomial touches or crosses the x-axis.
• These points help in understanding the graph's behavior and structure.

Often, to discover the zeros, methods like factorization or synthetic division are employed. In simpler polynomials, these methods help simplify the equation to easily find values of \(x\) that fulfill the zero-condition.

Factorization

Factoring a polynomial means expressing it as a product of its linear components. This process helps reveal the zeros of the function in a more straightforward manner.
For the polynomial \(f(x) = x^3 + 10x^2 + 33x + 34\), factorization shows it can be written as \((x+1)(x+2)(x+17) = 0\).

• Each component of the factored form, \((x + a)\), corresponds to potential solutions for the function when set to zero.
• By solving \((x+1)=0\), \((x+2)=0\), and \((x+17)=0\), we find the zeros \(x = -1, -2, \text{ and } -17\).

Factorization not only helps in solving polynomials more efficiently but also simplifies verification of calculations. When a polynomial is expressed in its factored form, identifying solutions is direct and avoids complex algebraic manipulation.

x-intercepts

In mathematics, especially in graphing, x-intercepts are crucial points where the graph of a function intersects the x-axis. These intercepts occur at the zeros of the function. For the polynomial \(f(x)\), the x-intercepts are calculated from the factors obtained during factorization.
For the given example, setting \((x+1)(x+2)(x+17)=0\), the x-intercepts are found to be:

• \(x = -1\)
• \(x = -2\)
• \(x = -17\)

Each x-intercept represents a point where the polynomial equals zero and crosses or touches the x-axis. These points are essential for sketching the function and understanding its general shape and direction.

Graphing Utility

Using a graphing utility or tool is an excellent way to visually verify the solutions obtained from algebraic manipulations. Graphing tools plot the function across various values of \(x\) and allow us to see where it intersects the x-axis.
When we graph \(f(x) = x^3 + 10x^2 + 33x + 34\), we should see the curve crossing the x-axis at the points \(x = -1, -2, \text{ and } -17\).

• The graph provides a visual confirmation that the calculated zeros are indeed the x-intercepts of the polynomial.
• It also helps verify that there are no additional points where the graph meets the x-axis.

Graphing utilities enhance understanding by translating numerical and algebraic solutions into a visual format, ensuring a robust verification method. They are fundamental tools both for checking work and for educational purposes.