Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$g(x)=x^{2}+10 x+23$$

Step-by-Step Solution

Answer

The zeros of the function \(g(x) = x^2 + 10x + 23\) are \(-5 - i\) and \(-5 + i\). The polynomial can be re-written as \(g(x) = (x + 5 - i)(x + 5 + i)\)

Step 1: Set equation to zero

Firstly, for finding the zeros of function \(g(x)\), it is required that the equation is set to zero. \[g(x) = x^{2}+10 x+23 = 0\]

Step 2: Apply Quadratic Formula

Now solve the equation for \(x\) applying the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here, \(a = 1\), \(b = 10\) and \(c = 23\).

Step 3: Substitute and Simplify

Putting the values into the quadratic formula we get \[x = \frac{-10 \pm \sqrt{10^2 - 4(23)}}{2}\] Simplifying it further gives \[x = -5 \pm \sqrt{1}\] As the term under the square root is negative, this indicates that we'll have complex roots.

Step 4: The Zeros of the function

Subtracting and adding \(1\) to \(-5\), we obtain the zeros of the function which are \(-5 - i\) and \(-5 + i\).

Step 5: Rewrite the polynomial

The polynomial can be written as product of linear factors: \(g(x) = (x + 5 - i)(x + 5 + i)\)

Step 6: Verification on graphing tool

Verify the calculated roots graphically using any graphing tool. The graph of the function \(g(x)=x^{2}+10 x+23\) would not intersect the x-axis, as the function has complex roots.

Key Concepts

Quadratic Formula

The quadratic formula is the go-to tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). It allows us to find the roots by using the coefficients \(a\), \(b\), and \(c\). Here’s the equation itself:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

When you plug in the values into this formula, you are essentially solving for \(x\), revealing the points where the parabola, represented by your quadratic equation, will intersect the x-axis.
The plus-minus symbol (\(\pm\)) indicates that a parabola's shape can intersect the x-axis in two places, have one point of contact, or not meet it at all. In our case, solving \(x^2 + 10x + 23 = 0\) helped determine the nature of the roots, which were not real in this instance.

Complex Roots

Sometimes the expression inside the square root of the quadratic formula, known as the discriminant, is negative. This results in complex numbers as roots because the square root of a negative number isn't a real number. Instead, it produces imaginary roots where the imaginary unit \(i\) is utilized, defined as \(\sqrt{-1}\).
For our equation, \(x^2 + 10x + 23\), the discriminant was negative. Specifically, the equation inside the square root became \(10^2 - 4 \times 23 = -12\). This negative result led to complex roots, \(-5 + i\) and \(-5 - i\).
Such roots indicate that the graph of the polynomial does not intersect the x-axis, confirming that the zeros are not real, but complex. They can be visualized as an extension of the number line into a complex plane.

Polynomial Factorization

Once we have the complex roots, we can write the polynomial as a product of linear factors. Linear factors are expressions that can be multiplied to give the original polynomial when expanded. These factors reflect both the real and complex solutions of the polynomial.
With the zeros \(-5 + i\) and \(-5 - i\), the polynomial \(g(x) = x^2 + 10x + 23\) can be expressed as \((x + 5 - i)(x + 5 + i)\). This factorization turns the original equation into a form that easily demonstrates the roots.
Recognizing the linear factors helps understand how polynomials are built and broken down by factoring, which is crucial in solving and simplifying complex problems in algebra.

Graphing Utility

Using a graphing utility is a beneficial method for verifying roots, especially when they are complex. Graphing tools allow students to interact with functions visually, offering a representation of a polynomial’s behavior.

• In cases with real roots, the x-axis will intersect at specific points.
• For complex roots, the graph might show some abstract behavior, but it will notably not touch the x-axis.

For the function \(g(x) = x^2 + 10x + 23\), a graphing tool confirms that it doesn’t cross the x-axis, thus agreeing with our earlier conclusion about the presence of complex roots, \(-5 \pm i\).
This visualization helps solidify understanding by allowing students to see the results of their calculations and to appreciate how solutions can appear outside of real-number systems.