Question

What are the zeros of the function \(y=(x+4)(x-7) ?\) \(\begin{array}{lll}\text { (A) } & -2 \text { and } 4 & \text { OR }\end{array}\) (B) 2 and \(-4\)

Short answer

The zeros are \(-4\) and \(7\).

Step-by-step solution

Set the function equal to zero

To find the zeros of the function, start by setting the function equal to zero: \( (x+4)(x-7) = 0 \). This equation shows where the function crosses the x-axis.

Apply the zero product property

If a product of factors equals zero, then at least one of the factors must be zero. Use this property by setting each factor equal to zero separately: 1) \( x+4 = 0 \)2) \( x-7 = 0 \).

Solve each equation for x

Solve the equations from the previous step to find the values of x:1) For \( x+4 = 0 \), subtract 4 from both sides to get \( x = -4 \).2) For \( x-7 = 0 \), add 7 to both sides to get \( x = 7 \).

Key concepts

Understanding the Zeros of a Function

To find the zeros of a function is to determine where the function intersects the x-axis. This happens when the output of the function, usually denoted by "y", equals zero. For example, given the function \( y = (x+4)(x-7) \), the zeros are the solutions to the equation \( (x+4)(x-7) = 0 \). Finding these points helps us understand key characteristics of the function, such as its behavior and roots.
These zeros are found by setting the expression equal to zero because the function graphs a curve on the coordinate plane, and zeros are precisely where it "touches" or crosses the x-axis. In simpler terms, they are the x-values for which the function's output is zero, making them vital for graphing and analyzing polynomial functions.
To summarize:

• Zeros are where \( y = 0 \).
• They are critical points on the graph.
• Understanding them aids in graphing the function.

Zero Product Property Explained

The Zero Product Property is a fundamental concept in algebra, especially when solving quadratic equations like \( (x+4)(x-7) = 0 \). This property states that if the product of two numbers (or expressions) is zero, then at least one of the factors must be zero. This comes from the principle that zero times any number is zero.
Applying this property is straightforward: for any equation of the form \( (a)(b) = 0 \), either \( a = 0 \) or \( b = 0 \). In our example, we break it down to:

• The first factor: \( x+4 = 0 \)
• The second factor: \( x-7 = 0 \)

Finding solutions independently for each factor allows us to determine all possible zeros of the original function. The beauty of the Zero Product Property lies in its simplicity and its powerful application to polynomial equations.
In essence, understanding this property:

• Enables quick and efficient problem-solving.
• Is crucial for isolating equations by their factors.
• Provides a foundation for solving more complex algebraic equations.

Solving Quadratic Equations

Solving quadratic equations often involves finding the zeros of the function. Quadratic equations take the general form \( ax^2 + bx + c = 0 \). However, when they're in factored form, like \( (x+4)(x-7) = 0 \), they become easier to solve using properties like the Zero Product Property.
The process begins by setting the equation equal to zero, aligning it with the purpose of finding the function's zeros. If it's not already factored, use techniques such as factoring, completing the square, or applying the quadratic formula to isolate solutions.
In our simple case:

• Recognize the factored form: \( (x+4)(x-7) \).
• Set each factor to zero: \( x+4=0 \) and \( x-7=0 \).
• Solve these equations to find \( x=-4 \) and \( x=7 \).

Once these roots are identified, they provide not only the zeros but often also intercepts for the graph. Quadratic equations, when solved, reveal much about the function’s shape and direction.
This approach:

• Builds a solid foundation for understanding polynomial equations.
• Shows the relationship between algebraic expressions and their graphical representations.
• Is imperative for high-level mathematics and applications.