What are the zeros of the function $y=(x-4)\left(x^2-12 x+36\right)$?
The function is given by \(y = (x - 4)(x^2 - 12x + 36)\). To find the zeros, we need to determine the values of \(x\) that satisfy the equation \(y = 0\). This leads to the equation \((x - 4)(x^2 - 12x + 36) = 0\). According to the zero product property, the product of two factors is zero if and only if at least one of the factors is equal to zero. Therefore, we set each factor equal to zero: \(x - 4 = 0\) and \(x^2 - 12x + 36 = 0\).
The first part of the equation is \(x - 4 = 0\), which is a simple linear equation. Adding 4 to both sides gives the first zero: \(x = 4\). The second part of the equation is the quadratic \(x^2 - 12x + 36 = 0\). We can factor this expression by identifying it as a perfect square trinomial. Since the square root of 36 is 6 and \(2 \times 6 = 12\), the expression can be written as \((x - 6)^2 = 0\). Taking the square root of both sides gives \(x - 6 = 0\), which results in the second zero: \(x = 6\).
Combining the results from the two parts, the zeros of the function are \(4\) and \(6\). We can see that this matches the second option provided in the problem description. Therefore, the correct zeros are \(4\) and \(6\).
Understanding how to find the zeros of a function is essential in various real-life scenarios, such as in engineering and economics. For example, if you are modeling the profit of a company as a function of the price of a product, the zeros of that function represent the "break-even points"—the prices at which the profit is exactly zero. Knowing these points allows businesses to understand their operating range and make informed decisions about pricing and production levels. By solving equations like \((x - 4)(x^2 - 12x + 36) = 0\), we can identify these critical boundary values.