Unlocking Zeros: Factoring Polynomials in Precalculus

PreAlgebra Grades High School 2:45 Video
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Lesson Description

Master the concept of zeros of polynomial functions by factoring. Learn to identify zeros algebraically and graphically, connecting solutions to x-intercepts. We'll explore real and imaginary zeros and their graphical implications.

Video Resource

Zeros - How to Find Using Factoring

Mario's Math Tutoring

Duration: 2:45
Watch on YouTube

Key Concepts

  • Zeros of a function (roots, solutions, x-intercepts)
  • Factoring polynomials
  • The relationship between real zeros and x-intercepts
  • Imaginary zeros and their graphical implications

Learning Objectives

  • Define zeros of a function and their graphical representation.
  • Find zeros of quadratic functions by factoring.
  • Explain the difference between real and imaginary zeros and how they relate to the graph of a function.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a zero of a function. Emphasize the multiple names for zeros (roots, solutions, x-intercepts). Briefly discuss the graphical interpretation of zeros as x-intercepts. Show the video from 0:04-0:21.
  • Factoring Example 1 (7 mins)
    Work through the first example from the video (0:51-1:24) together. Pause the video after Mario introduces the problem. Have students attempt to factor the quadratic. Then, resume the video and discuss the solution, emphasizing setting each factor equal to zero.
  • Factoring Example 2 (7 mins)
    Similar to Example 1, guide students through the second example in the video (1:24-1:51). Have students factor x^2 - 9 and solve for the zeros. Discuss the connection to the difference of squares factoring pattern.
  • Imaginary Zeros Example (8 mins)
    Play the third example in the video (1:51-end). Discuss how obtaining imaginary solutions means there are no real x-intercepts. Relate this to the graph of the quadratic and how it does not cross the x-axis.
  • Practice and Wrap-up (8 mins)
    Provide students with practice problems to solve on their own, similar to the examples in the video. Circulate to provide assistance. Summarize the key points of the lesson, including the definition of zeros, how to find them by factoring, and the graphical significance of real and imaginary zeros.

Interactive Exercises

  • Factoring Relay Race
    Divide the class into teams. Each team receives a set of quadratic equations. Team members take turns factoring the equations and solving for the zeros. The team that correctly solves all equations first wins.
  • Graphical Analysis
    Provide students with graphs of quadratic functions and ask them to estimate the real zeros from the graph. Then, have them find the zeros algebraically and compare their results.

Discussion Questions

  • What are the different names for zeros of a function?
  • How can you determine the number of real zeros a quadratic function has based on its graph?
  • How do imaginary solutions arise when finding zeros, and what does this tell us about the graph?

Skills Developed

  • Factoring polynomials
  • Solving quadratic equations
  • Connecting algebraic solutions to graphical representations
  • Critical thinking and problem-solving

Multiple Choice Questions

Question 1:

Which of the following is another term for the zeros of a function?

Correct Answer: Roots

Question 2:

What does it mean graphically when a quadratic function has two real zeros?

Correct Answer: The parabola intersects the x-axis at two points.

Question 3:

If a quadratic function has no real zeros, what type of zeros does it have?

Correct Answer: Imaginary

Question 4:

What is the first step in finding the zeros of f(x) = x^2 - 5x + 6 by factoring?

Correct Answer: Factor the quadratic expression

Question 5:

What are the zeros of the function f(x) = (x - 2)(x + 3)?

Correct Answer: 2 and -3

Question 6:

Which of the following functions does NOT have any real zeros?

Correct Answer: f(x) = x^2 + 4

Question 7:

What is the zero of f(x) = x - 7?

Correct Answer: 7

Question 8:

If you factor a quadratic and get (x + a)(x + b) = 0, what are the solutions for x?

Correct Answer: -a and -b

Question 9:

The graph of a quadratic function with imaginary zeros will...

Correct Answer: Not intersect the x-axis.

Question 10:

Which factoring method is used to factor x^2 - 9?

Correct Answer: Difference of Squares

Fill in the Blank Questions

Question 1:

The zeros of a function are also known as its _____ or solutions.

Correct Answer: roots

Question 2:

Graphically, real zeros of a function represent the _____ of the function.

Correct Answer: x-intercepts

Question 3:

If a quadratic equation has imaginary roots, its graph does not cross the _____

Correct Answer: x-axis

Question 4:

To find the zeros of a function by factoring, you must first set the function equal to _____.

Correct Answer: zero

Question 5:

The factored form of x^2 - 4 is (x - 2)(x + _____) .

Correct Answer: 2

Question 6:

If the discriminant (b^2 - 4ac) of a quadratic equation is negative, the quadratic equation has _____ zeros.

Correct Answer: imaginary

Question 7:

When factoring, we use the _____ property to set each factor equal to zero.

Correct Answer: zero product

Question 8:

The zero of f(x) = x is _____

Correct Answer: 0

Question 9:

The function f(x) = x^2 + 1 has _____ real zeros.

Correct Answer: no

Question 10:

Factoring is the _____ of expanding.

Correct Answer: opposite

Educational Standards

PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features.

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