Question

Find the zero(s) of the function. (See Example 4.) \(g(x)=x^2+22 x+121\)

Short answer

The zero of the function \(g(x) = x^2 +22x + 121\) is \(x = -11\).

Step-by-step solution

Identify the coefficients

In the given equation \(g(x)=x^2+22x+121\), the coefficients can be identified as \(a = 1\), \(b = 22\), \(c = 121\).

Apply the Quadratic formula

Substitute the values of \(a\), \(b\) and \(c\) into the quadratic formula \(x = {-b \pm \sqrt{b^2-4ac}}/{2a}\). This gives us \(x = {-22 \pm \sqrt{{22}^2-4*1*{121}}}/{2*1}\)

Simplify the above equation

Solving further, we get \(x = {-22 \pm \sqrt{484-484}}/{2}\). This simplifies to \(x = {-22 \pm 0}/2\) which further gives us \(x = -11\).

Key concepts

Zero of a Function

A zero of a function, also known as a root, is a value of the variable that makes the function equal to zero. In simpler terms, when you plug in this value into the function, the result is zero. For quadratic functions like \(g(x)=x^2+22x+121\), finding the function's zero involves solving the equation \(g(x)=0\). This means you're trying to find the value of \(x\) that makes \(g(x)=0\).

The zeros of a function are crucial because they show where the graph of the function intersects the x-axis in a coordinate plane. In our example, the function has a single zero at \(x = -11\). When plotted, the vertex of the parabola will just touch the x-axis at this point, indicating it is a perfect square trinomial. Finding zeros helps us analyze and understand the behavior of the quadratic function more comprehensively.

Quadratic Formula

The quadratic formula is a universal method for finding the zeros of any quadratic equation. A quadratic equation is typically in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. The quadratic formula is given by:

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

When using the quadratic formula, you substitute the coefficients from your quadratic equation into this formula to solve for \(x\).

For the equation \(g(x)=x^2+22x+121\), by substituting \(a=1\), \(b=22\), and \(c=121\) into the formula, the solution emerges. After computation, it simplifies to indicate that the quadratic has one repeated root at \(x = -11\).

This formula is invaluable because it can be used universally for any quadratic situation, regardless of the coefficients, providing either two real, one real, or two complex roots based on the discriminant (the value under the square root in the formula).

Polynomial Roots

Polynomial roots are the values that make the polynomial equal to zero, similar to the zeros in any function. For quadratic polynomials, these are most often referred to simply as roots or solutions. Understanding polynomial roots helps in graphing the polynomial and analyzing its properties, like intercepts and turning points.

In the case of \(g(x) = x^2 + 22x + 121\), the root is \(x = -11\). This single root is also called a double root or a repeated root because the value satisfies the equation twice as it represents both values of \(x\) when the expression is factored.

• Factoring the polynomial can further illustrate this: \((x+11)^2 = 0\).
• Setting each factor equal to zero gives back the double root: \(x+11=0\), solving gives \(x=-11\).

Roots are essential as they define key points on the graph of the polynomial, determining where the curve touches or crosses the x-axis. Understanding them allows for predicting the behavior and shape of the curve.