Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to find the real zeros of the function, and then (b) use the real zeros to find the exact values of the imaginary zeros. $$f(x)=x^{3}+4 x^{2}+14 x+20$$

Step-by-Step Solution

Answer

The function f(x) has one real root at x = -2 and two imaginary roots at x = -1 - 3i and x = -1 + 3i.

Step 1: Use a graphing utility to plot the function

Plot the cubic function \(f(x)=x^{3}+4 x^{2}+14 x+20\) using a graphing tool. The real zeros are essentially the x-coordinate of the points where the curve intersects the x-axis.

Step 2: Identify the real zeros from the graph

From the graph, it is noticed that the curve touches X-axis at \(x=-2\). So real root of the function is -2.

Step 3: Divide the given polynomial by \((x - realzero)\) using synthetic division

Now, use synthetic division to divide the given polynomial by \((x+2)\). This will reduce the cubic polynomial to a quadratic polynomial that can be solved easily. The quotient after synthetic division is \(x^{2}+2x+10\).

Step 4: Find the imaginary zeros

Now solve the derived quadratic equation \(x^{2}+2x+10 = 0\). We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). If the value inside the square root is negative, the roots are imaginary( complex roots). Substituting \(a=1, b=2, c=10 \) in the equation we get \(x= -1 \pm 3i\).

Step 5: State the final answer

The result shows that the cubic polynomial \(f(x)=x^{3}+4 x^{2}+14 x+20\) has one real root \(x=-2\) and two complex roots \(x= -1 + 3i\) and \(x= -1 -3i\).

Key Concepts

Graphing Utility

A graphing utility is an indispensable tool when it comes to understanding functions, especially polynomial ones. Think of it as a special calculator that not only handles numbers but can also draw graphs.
By inputting the function equation, such as \(f(x)=x^{3}+4x^{2}+14x+20\), the graphing utility provides a visual representation. This function's curve is plotted on the coordinate plane, enabling us to see where it intersects the x-axis. These intersection points are the real zeros, or roots, of the function.
In our case, the curve touches the x-axis at \(x = -2\). This immediate visualization allows students to confirm analytically derived or suspected roots, making the learning experience richer and more interactive.

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, particularly useful when you've identified a real zero of the polynomial. It's a bit like long division but designed specifically for polynomials, and it's faster.
Let's highlight how synthetic division works with our polynomial \(f(x) = x^{3}+4x^{2}+14x+20\) and its real zero \(x = -2\). First, translate the zero as \(x+2\), then arrange the coefficients of the polynomial in descending order: [1, 4, 14, 20].
You perform synthetic division by following these steps:

• Write down the zero \(x = -2\) outside the division bracket.
• Bring down the leading coefficient \(1\) to the bottom row.
• Multiply \(-2\) by \(1\) and write the result below the next coefficient \(4\).
• Add this result to \(4\), place the sum \(2\) in the row below, and repeat for the rest.

This process yields the reduced polynomial \(x^{2}+2x+10\), making it significantly easier to solve.

Quadratic Formula

When you have a quadratic equation like \(ax^{2} + bx + c = 0\), the quadratic formula is your go-to tool for finding the roots. The formula is:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
This formula solves for \(x\), giving either real or complex roots, depending on the discriminant \(b^{2} - 4ac\).
In our reduced case from synthetic division, \(x^{2} + 2x + 10 = 0\):

• Identify \(a = 1\), \(b = 2\), and \(c = 10\).
• Plug these values into the formula to find \(x = \frac{-2 \pm \sqrt{2^{2} - 4 \times 1 \times 10}}{2 \times 1}\).

Since the discriminant \(2^{2} - 40 = -36\) is negative, the roots are not real numbers but complex numbers, \(x = -1 \pm 3i\).

Complex Numbers

Complex numbers are an extension of real numbers and are crucial when dealing with polynomials where the discriminant is negative, leading to imaginary roots. A complex number comprises a real part and an imaginary part and is expressed in the form \(a + bi\), where \(i = \sqrt{-1}\).
In our case with the quadratic \(x^{2} + 2x + 10\), the roots were found to be complex: \(-1 + 3i\) and \(-1 - 3i\). The term \(3i\) indicates that the square root involved is not real, but rather involves \(i\), the unit of imaginary numbers.
Complex numbers are essential in various fields such as electronics and quantum physics, as they help in understanding phenomena that are not easily explained by real numbers alone. They add a robust dimension to solving polynomial equations, especially when the behavior of graphs doesn't cross real number solutions on the x-axis.