Zeros Found! Master Functions Algebraically (Easy Guide)

The concept of a zero of a function, a fundamental element in algebra, represents the x-value where the function intersects the x-axis. Khan Academy provides valuable resources, including tutorials and practice problems, to deepen your understanding of function zeros. Solving quadratic equations is often a necessary step in how to find the zeros of a function algebraically, especially when dealing with polynomial functions. Understanding the Factor Theorem can simplify the process, allowing you to determine if a given value is a zero of a polynomial. This guide will provide you with an easy-to-follow approach on how to find the zeros of a function algebraically.

Image taken from the YouTube channel Brian McLogan , from the video titled How To Find the Zeros of The Function .

Zeros Found! Master Functions Algebraically (Easy Guide)

This guide breaks down how to find the zeros of a function algebraically, making a potentially challenging concept understandable. We'll explore various techniques and provide clear examples to help you master this essential skill.

What are Zeros of a Function?

Before diving into how to find the zeros of a function algebraically, it's crucial to understand what they are.

• Definition: The zeros of a function are the values of x that make the function equal to zero. In other words, they are the solutions to the equation f(x) = 0.
• Graphical Representation: On a graph, the zeros are the x-intercepts – the points where the function's graph crosses the x-axis.
• Importance: Finding zeros is fundamental in many areas of mathematics and its applications, including solving equations, optimization problems, and modeling real-world phenomena.

Techniques for Finding Zeros Algebraically

This section covers several methods you can use to algebraically determine the zeros of a function.

Factoring

Factoring is one of the most straightforward techniques when applicable.

1. Set the Function to Zero: Begin by setting the function, f(x), equal to zero.
2. Factor the Expression: Factor the resulting expression completely.
3. Set Each Factor to Zero: Set each individual factor equal to zero.
4. Solve for x: Solve each of the resulting equations for x. The solutions are the zeros of the function.

Example:

Find the zeros of f(x) = x² - 4x + 3.

• x² - 4x + 3 = 0
• (x - 3)(x - 1) = 0
• x - 3 = 0 or x - 1 = 0
• x = 3 or x = 1

Therefore, the zeros of the function are x = 3 and x = 1.

Using the Quadratic Formula

The quadratic formula is used specifically for finding the zeros of quadratic functions (functions of the form f(x) = ax² + bx + c).

1. Identify a, b, and c: Determine the values of a, b, and c in the quadratic equation.

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2. Apply the Formula: Substitute the values of a, b, and c into the quadratic formula:

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   x = (-b ± √(b² - 4ac)) / 2a

3. Simplify: Simplify the expression to find the two possible values of x, which are the zeros of the function.

Example:

Find the zeros of f(x) = 2x² + 5x - 3.

• a = 2, b = 5, c = -3
• x = (-5 ± √(5² - 4 2 -3)) / (2 2)*
• x = (-5 ± √(25 + 24)) / 4
• x = (-5 ± √49) / 4

• x = (-5 ± 7) / 4

  This gives us two solutions:

  • x = (-5 + 7) / 4 = 2 / 4 = 1/2
  • x = (-5 - 7) / 4 = -12 / 4 = -3

Therefore, the zeros of the function are x = 1/2 and x = -3.

Isolating the Variable (for Simpler Functions)

For certain functions, especially those with only one instance of x, you can isolate the variable.

1. Set the Function to Zero: As always, begin by setting f(x) = 0.
2. Isolate x: Use algebraic operations to isolate x on one side of the equation.

Example:

Find the zero of f(x) = 3x + 6.

• 3x + 6 = 0
• 3x = -6
• x = -2

Therefore, the zero of the function is x = -2.

Using the Square Root Property

When dealing with equations where x is squared and can be isolated, the square root property is a useful method.

1. Isolate the Squared Term: Isolate the term containing x².
2. Take the Square Root of Both Sides: Take the square root of both sides of the equation. Remember to include both the positive and negative square roots.
3. Solve for x: Solve for x.

Example:

Find the zeros of f(x) = x² - 9.

• x² - 9 = 0
• x² = 9
• x = ±√9
• x = ±3

Therefore, the zeros of the function are x = 3 and x = -3.

Dealing with More Complex Functions

Sometimes, finding the zeros algebraically can be challenging or impossible using simple techniques. In these cases:

• Advanced Factoring Techniques: Consider advanced factoring methods like factoring by grouping.
• Rational Root Theorem: If the function is a polynomial, the Rational Root Theorem can help identify potential rational roots.
• Numerical Methods: For very complex functions, numerical methods like Newton's method may be needed to approximate the zeros. These methods are often implemented using calculators or computer software.

A Table Summarizing the Methods

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