pre calculus factoring worksheet

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• Extrema, intervals of increase and decrease
• Power functions
• Average rates of change
• Transformations of graphs
• Piecewise functions
• Graphs, real zeros, and end behavior
• Dividing polynomial functions
• The Remainder Theorem and bounds of real zeros
• Writing polynomial functions and conjugate roots
• Complex zeros & Fundamental Theorem of Algebra
• Graphs of rational functions
• Rational equations
• Polynomial inequalities
• Rational inequalities
• Graphing exponential functions
• Exponential equations not requiring logarithms
• Exponents and logarithms
• Evaluating logarithms
• Logarithms and exponents as inverses
• Properties of logarithms
• Writing logs in terms of others
• Exponential equations requiring logarithms
• Logarithmic equations, simple
• Logarithmic equations, hard
• Graphing logarithmic functions
• Compound interest
• Angles and angle measure
• Right triangle trigonometry
• Trig functions of any angle
• Graphing trig functions
• Simple trig equations
• Inverse trig functions
• Fundamental identities
• Equations with factoring and fundamental identities
• Sum and Difference Identities
• Multiple-Angle Identities
• Product-to-Sum Identities
• Equations and Multiple-Angle Identities
• The Law of Sines
• The Law of Cosines
• Parametric equations
• Projectile motion
• Polar coordinates
• Graphs of polar equations
• Polar and rectangular forms of equations
• Polar forms of conic section
• Complex numbers and polar form
• Vector basics
• Vector operations
• Dot products
• 3D vector basics
• 3D vector operations
• Vector cross products
• Matrix operations
• Matrix inverses and determinants
• Matrix equations
• Cramer's Rule
• Multivariable linear systems and row operations
• Partial fraction decomposition
• Rotations of conic sections
• Sample spaces & Fundamental Counting Principle
• Permutations
• Combinations
• Permutations vs combinations
• The Binomial Theorem
• Mathematical induction
• Independent and dependent events
• Mutually exclusive events
• Probability with combinatorics
• Probability with combinatorics - binomial
• General sequences and series
• Arithmetic sequences and series
• Geometric sequences and series
• Power series
• Limits by direct evaluation
• Limits at kinks and jumps
• Limits at removable discontinuities
• Limits at essential discontinuities
• Limits at infinity
• Definition of the derivative
• Instantaneous rates of change
• Power rule for differentiation
• Motion along a line
• Approximating area under a curve
• Area under a curve by limit of sums
• Indefinite integrals

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Chapter 5.5: Factoring Special Products

Now transition from multiplying special products to factoring special products. If you can recognize them, you can save a lot of time. The following is a list of these special products (note that a 2 + b 2 cannot be factored):

The challenge is therefore in recognizing the special product.

Factor each of the following polynomials.

Answers to odd questions

Pre-Calculus Copyright © 2022 by St. Clair College is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Precalculus Examples

• Finding the GCF of a Polynomial
• Factoring Out Greatest Common Factor (GCF)
• Identifying the Common Factors
• Cancelling the Common Factors
• Factoring Trinomials
• Trinomial Squares
• Factoring Using Any Method
• Factoring a Difference of Squares
• Factoring a Sum of Cubes
• Factoring by Grouping
• Factoring a Difference of Cubes
• Determine if an Expression is a Factor
• Determining if Factor Using Synthetic Division
• Find the Factors Using the Factor Theorem
• Determining if Polynomial is Prime
• Determining if the Polynomial is a Perfect Square
• Expand using the Binomial Theorem
• Finding All Integers k Such That the Trinomial Can Be Factored
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Precalculus MAT 206–0801

Spring 2020 BMCC Prof. Dr. Ivan Retamoso

Relations and Functions

Evaluations of Functions

The Horizontal Line Test

Finding the Domain of a Function

Finding the Domain and Range of a Function from its Graph

Piecewise Functions

Average Rate of Change of a Function

Increasing and Decreasing Functions

Relative Maxima or Minima

Arithmetic Operations with Functions

Compositions of Functions

Transformation of Functions

Absolute Value Functions

Inverse Functions

Linear Functions

Graph of Linear Functions

Modeling with Linear Functions

Operations with Complex Numbers

Vertex and Axis of Symmetry of a Parabola

Minimum or Maximum of a Parabola

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

The Reminder and Factor Theorem

The Rational Zero Theorem

Zeros of a Polynomial Function

Rational Functions

Graphing Rational Functions

Slant Asymptote

Exponential Functions

Logarithmic Functions

Properties of Logarithms

Exponential and Logarithmic Equations

Radian and Degree Measure

Right Triangle Trigonometry

Trigonometric Functions of any Angle

The Unit Circle

Graph of Trigonometric Functions

Inverse Trigonometric Functions

Sum and Difference Identities

Double-Angle and Half-Angle Formulas

Product to Sum and Sum to Product Formulas

Solving Trigonometric Equations

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Precalculus

Unit 1: composite and inverse functions, unit 2: trigonometry, unit 3: complex numbers, unit 4: rational functions, unit 5: conic sections, unit 6: vectors, unit 7: matrices, unit 8: probability and combinatorics, unit 9: series, unit 10: limits and continuity.

Pre-Calculus 11

BC Pre-Calculus 11 Youtube Playlist

Click  here  for a concise playlist on BC Pre-Calculus 11 practice problems. Each math video is about 5 minutes long and can be used as lesson examples or for reviewing the course. Subscribe to the channel if you find these videos useful!

Pre-Calculus 11 Topics

• Types of Numbers – hunkim.com/11a
• Powers – hunkim.com/11b
• Radicals – hunkim.com/11c
• Trigonometry – hunkim.com/11d
• Polynomial Factoring – hunkim.com/11e
• Quadratic Functions and Equations – hunkim.com/11f
• Linear and Quadratic Inequalities – hunkim.com/11g
• Rational Expressions – hunkim.com/11h
• Financial Literacy – hunkim.com/11i

Pre-Calculus 11 Topic A: Types of Numbers

Types of Numbers is the first topic of the BC Pre-Calculus 11 curriculum. The symbol for integers \mathbb{Z} is based on the German word “Zahlen.” BC students do not have to worry about complex numbers but BC math students who are interested in studying in the US should learn that i=\sqrt{-1} .

• Real number classification
• Types of Numbers 1
• Types of Numbers 1 (solutions)

Pre-Calculus 11 Topic B: Powers

Powers is another topic in the BC Pre-Calculus 11 curriculum. Although you may have learned about exponent laws since Grade 9, this is the first year that BC math students fully learn the topic of powers which include negative exponents and fractional exponents.

• Positive and negative rational exponents
• Exponent laws
• Evaluation using order of operations
• Numerical and variable bases
• Powers 1 (solutions)

Pre-Calculus 11 Topic C: Radicals

Radicals is a math topic that has a connection with exponents. A fractional exponent can be rewritten as a radical. This year in Pre-Calculus 11 we will focus on simplifying radicals and working with radical equations. Next year you will learn more about graphing and transforming radical functions.

• Simplifying radicals
• Ordering a set of irrational numbers
• Performing operations with radicals
• Solving simple (one radical only) equations algebraically and graphically
• Identifying domain restrictions and extraneous roots of radical equations
• Radicals 1 (solutions)

Pre-Calculus 11 Topic D: Trigonometry

Last year you learned about right-angle trigonometry: SOH CAH TOA. This year in Pre-Calculus 11 you will learn how to solve non right-angle triangles using the Sine Law and the Cosine Law. Do your best to understand this year’s trigonometry concepts because you will learn more about trigonometry next year. Visit hunkim.com/10 to review SOH CAH TOA Trigonometry.

• Use of sine and cosine laws to solve non-right triangles, including ambiguous cases
• Contextual and non-contextual problems
• Angles in standard position
• Special angles, as connected with the 30-60-90 and 45-45-90 triangles
• Unit circle
• Reference and co-terminal angles
• Terminal arm
• Trigonometric ratios
• Simple trigonometric equations
• Label the location of the four quadrants

Pre-Calculus 11 Topic E: Polynomial Factoring

We continue factoring this year in Pre-Calculus 11 because factoring is an important part of quadratic functions and equations. This year we factor quadratics and next year we factor higher degree polynomials. By factoring, we reveal the roots of a polynomial which are also known as x-intercepts or solutions.

• Greatest common factor of a polynomial
• Trinomials of the form ax^2+bx+c
• Difference of squares of the form a^2x^2-b^2y^2
• May extend to a(f(x))^2+b(f(x))+c and a^2(f(x))^2-b^2(f(x))^2
• Factor 2x^2-4x

Pre-Calculus 11 Topic F: Quadratic Functions and Equations

Quadratics is the largest math topic in Pre-Calculus 11. We can use our understanding of quadratics to determine the exact height of a basketball or find the optimal dimensions to maximize or minimize area.

• Identifying characteristics of graphs (including domain and range, intercepts, vertex, symmetry), multiple forms, function notation, extrema
• Exploring transformations
• Solving equations (ex. factoring, quadratic formula, completing the square, graphing, square root method)
• Connecting equation-solving strategies
• Connecting equations with functions
• Solving problems in context
• Sketch and label three points

Pre-Calculus 11 Topic G: Linear and Quadratic Inequalities

In Pre-Calculus 11, we extend our understanding of quadratics and lines by studying the meaning of the inequality symbol. You may be familiar with solving the equation x^2=9 . But did you know that x^2<9 is equivalent to -3<x<3 ?

• single variable (ex. 3x-7\leq-4 , x^2-5x+6>0 )
• domain and range restrictions from problems in situational contexts
• sign analysis: identifying intervals where a function is positive, negative, or zero
• symbolic notation for inequality statements, including interval notation
• Sketch y=3x+2

Pre-Calculus 11 Topic H: Rational Expressions

This year, in Pre-Calculus 11, we focus on simplifying rational expressions and introduce a few word problems. Next year you will better understand how to graph rational functions.

• Simplifying and applying operations to rational expressions
• Identifying non-permissible values
• Solving equations and identifying and extraneous roots
• What is a rational expression?

Pre-Calculus 11 Topic I: Financial Literacy

As a young person, you have the power to be wealthy even with a minimum wage job. By living within (or below) your means, you will have money left over to invest. If you invest $6 a day (less than the cost of some Starbucks drinks) for 50 years (at an annual interest rate of 10%) your investment will grow to be more than $3 million!

• compound interest
• introduction to investments / loans with regular payments, using technology
• buying / leasing
• The simple interest formula is A=P(1+rt) . If the annual interest rate is 10%, explain how a $1000 investment would grow over three years.
• Click here for more BC math practice problems.
• Click here for more math practice problems.

Radians and Degrees

When precalculus students learn about degrees, minutes, and seconds, they become fluent in converting between degrees and minutes, as well as converting between radians and degrees. They also learn how to find the area of sector or sector length. Precalculus is challenging for many students in general, and learning about radians and degrees is no […]

Trigonometric Functions on the Unit Circle

When precalculus students learn about trigonometric functions on the unit circle, they learn how to understand the standard position of an angle, coterminal angle, and reference angle and become fluent in finding the trigonometric ratios in a given point on a unit circle. Even though trigonometry can cause headaches for even the best of students, […]

Logarithmic Functions

In precalculus, students learn about functions, including logarithmic functions. They learn what logarithmic functions represent and acquire the necessary skills to evaluate such functions and find their domain and range. It’s no secret that logarithms can be challenging to master. Luckily, there are tons of tips that math teachers can use to help out students […]

Properties of Logarithms

When pre-calculus students learn about logarithmic functions, one of the most important lessons they come across is the properties of logarithms. This is because students can simplify and evaluate logarithms with the help of these properties. Even though log lessons may be challenging, math teachers can help make them more engaging and accessible by using […]

Matrix Operations

In precalculus, students learn about matrices and how to apply mathematical operations to them. Matrices have many useful applications, such as representing numerical data in a more compact form in tables, making it easier to manipulate it and find solutions to problems. Regardless of their usefulness, matrices and matrix operations can come across as confusing […]

Analyzing Graphs of Functions and Relations

Once pre-calculus students are confident in their knowledge of functions and relations, they move on to analyzing graphs of functions and relations (including identifying x and y-intercepts, zeros, symmetry, even and odd functions). Learning how to analyze graphs of functions and relations often comes across as tedious for many students. So one of the biggest […]

Power and Radical Functions

When learning about functions in precalculus, students familiarize themselves with what power and radical functions are, how to define and graph them, as well as how to solve equations that contain radicals. Units in precalculus are often seen as challenging, and power and radical functions are no exception to this. To help out with your […]

Polynomial Functions

In precalculus, students learn about functions. In lessons on polynomial functions, students first learn how to define polynomial functions and how to find the zeros as well as roots of polynomial functions. Even though precalculus lessons are often perceived as daunting for many students, they can be turned into a fun and easy learning experience […]

Teaching Functions in Precalculus

In lessons on functions (precalculus), math students learn what function notation is and become proficient in evaluating and solving functions, including piecewise functions, as well as interpreting statements that use function notation in terms of a context. Functions in precalculus may be challenging to grasp even for students who excel at math, and many math […]

Teaching Quadratic Functions and Equations

When students start with precalculus lessons, they learn how to graph quadratic functions and what the properties of quadratic graphs are. They also develop an understanding of quadratic equations and learn about the different ways of solving them. While entering the realm of pre-calculus may come across as scary for many students, there are many […]

At the bottom of this lesson there are Guided Notes, a Slide Show, and a Sets Worksheet to help you out with this teaching Sets to your students. Set Notation: It all starts with Set Notation. Students must master these symbols so that they can correctly analyze the problems they will be doing… Students need […]

Arithmetic Sequences and Series Worksheet

In all classrooms, there are students with different levels of ability. No matter what concept is being taught, presenting the basics to the class and then providing activities that individualize instruction meets the needs of more students and provides opportunities for student engagement and feeling of success. Writing lesson plans for such classroom interaction can […]

Polar Coordinates – Baby Talk

You’ve heard the saying “talk to me like I am three” when trying to explain or understand something complex. The approach to introducing polar coordinates must be made using that same philosophy. For students to understand how to graph polar coordinates and equations, these lessons must use a scaffolding method where students understand each piece […]

Introduction to Vectors – Navigating the Seas

Early navigators used math to chart their course and stay on it when dealing with factors such as wind and ocean current that affected their journeys. The system left something to be desired regarding accuracy. With the use of Introduction to vectors, your students can complete the activity that is linked below and gain not […]

How to Teach Ellipses and Circles

When students see a concept presented in more than one way, the chances of understanding increase. Many teachers have students draw ellipses and circles by attaching a loop of string to two tacks placed in corrugated cardboard. Students use a pencil to draw the ellipse by placing it at one of the tacks and moving […]

How to Teach Matrix Multiplication

The ‘Illuminations” Lessons from the NCTM website are an excellent resource for many mathematics concepts. Math teachers often take the ideas of others and tweak them to suit their level of comfort or match the activity to the ability level of students. One of the lessons entitled, Computer Animation, is a fun way for students […]

Verifying Trigonometric Identities – Commit and Toss

WARNING 🙂 You are going to let students throw paper wads or paper airplanes in your class. The activity is used for closure during the later portion of a class period. Anytime you can interject some play, the task is less complicated, and students are more open to discussion with their peers. Commit and Toss […]

Right Triangle Trigonometry – Get Triggy Wit It

Start your Right Triangle Trigonometry lesson with the YouTube video linked below. It is based on Will Smith’s rap song ‘Gettin’ Jiggy Wit It,’ entitled, ‘Get Triggy Wit It.’ It may not have a lot of trigonometry content, but it does introduce sine, cosine, and tangent and how they relate to a right triangle. If […]

Exponential Functions – One Grain of Rice

Read the book, One Grain of Rice, to your students. You will be amazed at how older students will pay attention when you read the book to them. It is a nice way to introduce the concept of Exponential Functions and start the class differently than the norm. One Grain of Rice Activity Give them […]

Rational Functions – NCTM Activity

When trying to motivate students in your math class, it’s helpful if they can see how it applies to the real world. Don’t you get tired of students asking, “When am I ever going to use this?” We know they need to know math for more reason than just to pass the tests, but they […]

Continuity, End Behavior, and Limits – If This Then That

End behaviors can be a fun lesson to teach. There are a lot of “if this then that” situations that arise.    For example, if the degree of the function is even then the arrows on the end of the graphs will face the same direction. Continuity Continuous Functions have: • no breaks, holes, or […]

Operations with Complex Numbers – Total Recall

The title of the lesson is scarier than the actual lesson itself. When I start this lesson I always explain to the students that they  should not be scared because of the word complex. Kids are kids and the word complex will have them concerned before you even start. I have found it very helpful […]