Problem 38 - 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. $$f(x)=x^{2}-12 x+34$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=x^{2}-12x+34\) are \(x=6 \pm \sqrt{2}\). The polynomial can be expressed as a product of linear factors as \(f(x) = (x - (6 + \sqrt{2}))(x - (6 - \sqrt{2}))\), and the \(x\)-intercepts are at \(x=6 \pm \sqrt{2}\).

Step 1: Find the zeros of the function

Zeros of the function are obtained when \(f(x)=0\). Hence, solve for \(x\) in the equation \(x^{2}-12x+34=0\). In order to solve this quadratic equation, the quadratic formula is used, which is \(x = [-b \pm \sqrt{(b^2-4ac)}]/2a\). Thus, using \(a=1\), \(b=-12\), and \(c=34\), we find that \(x= [12 \pm \sqrt{((-12)^2-4*1*34)}]/2*1\). This simplifies to \(x= [12 \pm \sqrt{(144-136)}]/2\). So, the zeros of the function are \(x=6 \pm \sqrt{2}\).

Step 2: Write the polynomial as a product of linear factors

The original polynomial function can be expressed as a product of its linear factors using the zeros obtained. As such, \(f(x) = (x - (6 + \sqrt{2}))(x - (6 - \sqrt{2}))\).

Step 3: Determine the x-intercepts of the graph

The \(x\)-intercepts of the graph of the function are the same as the zeros of the function. Therefore, based on the zeros obtained in step 1, the \(x\)-intercepts of the function are \(x=6 \pm \sqrt{2}\).

Step 4: Verify the results with a graphing utility

Using a graphing utility, plot the function \(f(x)=x^{2}-12x+34\) to verify that the \(x\)-intercepts at \(x=6 \pm \sqrt{2}\) are the only intercepts.

Key Concepts

Zeros of a Function

Zeros of a function are those values of the variable that make the function's output equal to zero. In other words, they are the solutions to the equation set by the function equal to zero. For a quadratic function, such as the one in our exercise, the zeros can be found using the quadratic formula:

• The function is given by: \[x^2 - 12x + 34 = 0\]
• Using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• Plugging in the values: \[a = 1, \, b = -12, \, c = 34\]
• Solving, we find: \[x = 6 \pm \sqrt{2}\]

These zeros are the points where the function intersects the x-axis. They are crucially important for understanding both the algebraic and graphical behavior of the function.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its linear factors. Finding these factors can simplify the analysis of the function. For quadratic functions, once you have found the zeros, writing the function in factored form is straightforward. Given our function:

• The zeros are: \[x = 6 + \sqrt{2}, \quad x = 6 - \sqrt{2}\]
• The function in factored form is: \[f(x) = (x - (6 + \sqrt{2}))(x - (6 - \sqrt{2}))\]

This form clearly shows the roots of the polynomial. Each linear factor corresponds to a root of the equation, offering an intuitive understanding of the function's behavior around those points.

Graphing Utility

Using a graphing utility helps verify the solutions graphically. This tool can visually confirm where the graph crosses the x-axis and ensure that all calculated zeros are correct. When you graph the function, \[f(x) = x^2 - 12x + 34\] you should:

• Observe the curve shows intercepts on the x-axis exactly at the zeros, \[x = 6 \pm \sqrt{2}\]
• Note that there are no additional intercepts, affirming our calculations.
• Use this visual tool as a double-check method to confirm your algebraic findings.

This visualization aids in comprehending how polynomial roots correlate with their graphical interpretation.

X-Intercepts

The x-intercepts of a function are the points where the graph crosses the x-axis. These occur at the zeros of the function. The x-intercepts provide meaningful data about the graph's layout and can clarify how the function behaves over the real number span. For the quadratic function \[f(x) = x^2 - 12x + 34\], the x-intercepts are precisely the zeros, \[x = 6 + \sqrt{2} \quad\text{and}\quad x = 6 - \sqrt{2}\].

• This means that at these points, the value of the function is zero.
• Understanding these intercepts assists in sketching the overall shape of the graph.
• They are also crucial in solving real-world problems where the intersection points imply specific solutions or outcomes based on certain conditions.

By analyzing x-intercepts, we can unravel significant insights into the function's structure and its potential applications.