Description
POLYNOMIALS WHOLE UNIT for class 10 and 11! From an introduction to the polynomials unit [vocabulary words such as monomial, binomial, trinomial, term, degree, leading coefficient, divisor, quotient, dividend, etc.], then progresses deeper into the polynomials unit for how to calculate multiplicity, roots/zeros, end behavior, and finally sketching graphs of polynomials with varying degree and multiplicity. Students also match polynomial equations and their corresponding graphs. How to factor the polynomial - ALL methods, and how to solve polynomial equations by factoring. Polynomial Long Division, Synthetic Division, Factor and Remainder Theorems. How to identify intervals of increasing/decreasing, and [extrema] local/absolute max and min values. What is the Rational Roots Theorem? When and why do we use it? Finding all possible rational roots, actual rational roots, and all zeros of a polynomial function. Lessons are unique because they review important foundational skills FIRST.
Well-thought-out examples that flow, and excellent polynomial worksheets that tie everything together. The lessons make important connections and notes that students often miss; for example, about the relationships between real/imaginary zeros and x-intercepts.
✫ THIS PACKAGE IS A TOTAL OF 297 QUALITY PAGES!
INCLUDED:
- Detailed step-by-step answer keys and notes to EVERYTHING
- 9 Full lessons [PDF & SmartBoard Versions]
- 15 Worksheets
- 5 Reference sheets
- 2 Tests with study guide
- Creative degree>2 word problems
SUGGESTED ORDER:
1) How to FACTOR THE POLYNOMIAL [all methods step-by-step]
2) Solve Polynomial Equations BY FACTORING (which methods do I use!?)
3) FREE: Worksheet - Solve Equations by Formula or Factoring & Graphing
4) Intro, End Behavior, Multiplicity, Roots, Graph
5) Intervals of Increasing/Decreasing and Extrema [Local/Absolute Max & Min]
6) LONG DIVISION, FACTOR/REMAINDER THEOREMS
7) SYNTHETIC Division, FACTOR/REMAINDER Theorems
9) Degree>2 Real-World Word Problems
10) Polynomials UNIT TESTS with Study Guide
DETAILED LIST OF TOPICS:
➤ Definition: The word "polynomial" is broken down to "poly" and "nomial" ... which means many terms.
➤ Vocabulary words with examples: monomial, binomial, trinomial, term, degree, leading coefficient, root/zero, multiplicity, end behavior.
➤ Introduces multiplicity of roots and provides examples of:
- Determining a root and its corresponding multiplicity by looking at a polynomial equation in factored form.
- Sketching the graph of a polynomial function in factored form using the rules of multiplicity.
- Identifying a possible equation in factored form by looking at a polynomial graph with various multiplicity.
➤ Reviews:
- Identifying the degree and leading coefficient.
- Converting the equation to factored form by using the following factoring skills: by grouping, difference of cubes formula, and difference of squares formula.
- Using factored-form to help in finding the x-intercepts.
- Using the quadratic formula to produce 2 complex roots.
- Sketching the graph of a function where one of the roots has a multiplicity of 2, and knowing that two of the roots will not be graphed since they are complex.
➤ Students learn that the degree of a polynomial (whether it's even or odd) as well as the sign of the leading coefficient tie into determining the End Behavior [using proper notation]. Students then use this information to sketch a polynomial graph.
➤ Students sketch the graphs of polynomials with varying degree, using all the pieces of degree, roots/zeros, multiplicity and end behavior to help them.
➤ Makes very important connections that many students miss, if not pointed out, such as: "If a cubic function has 1 real root and 2 imaginary roots, how many x-intercepts will the function have? Draw a general picture."
➤ Students continue making important connections by sketching the graphs of quadratic and cubic functions with different numbers of x-intercepts. For example, sketching the graph of a quadratic function with 1 x-intercept versus none.
➤ Factoring - all methods:
1) Factor using the Greatest Common Factor (GCF) method
2) Factor by Grouping (4 terms)
3) Factor Quadratics (3 terms) - Both cases where leading coefficient (a) is equal to 1, and not equal to 1
4) Use the Difference of Perfect Squares formula to factor
5) Use the Sum & Difference of Cubes formula to factor
6) Factor a polynomial completely
➤ Long & Synthetic Division of polynomials.
➤ Factor/Remainder Theorems: Students are asked to determine whether the binomial is a factor of the dividend (only when the remainder=0); and if it is, then students can convert the polynomial into factored form by multiplying the divisor and quotient together.
➤ What is the Rational Roots Theorem? When and why do we use it? Finding all possible rational roots, actual rational roots, and all zeros of a polynomial function. Introductory lesson which reviews important foundational skills FIRST, and learning to distinguish whether or not Rational Roots Theorem is necessary in solving a polynomial equation.
➤ Creating polynomial word problem that reviews:
1) Identifying the x and y intercept(s) from the polynomial graph; then asking students to think about what do these points mean in the context of this stock market problem?
2) Finding intervals of increase and decrease.
3) Identifying the extrema (local/absolute max. and min. values). It also asks students to think about which extrema does not make sense in this real-world scenario?
4) Finding the exact equation in factored-form
5) Using the exact equation to make a future prediction
➤ 2 tests:
- Rearranging terms in an expression to convert into Standard Form.
- Identifying the degree and number of terms in an expression.
- Simplifying expressions.
- Factoring expressions.
- Identifying possible & exact equations of a polynomial graph in factored form using x-intercepts and multiplicity.
- Identifying x-intercepts and multiplicity from an equation.
- Solving equations by factoring and using the quadratic formula in simplest radical form.
- Identifying key features of a graph:
- Lowest possible degree
- X-and-Y-intercepts
- Intervals of increasing/decreasing
- Extrema (absolute & local max/min values)
- Operations (add/subtract/multiply and dividing using Long & Synthetic division)
Polynomials Full Unit | Lessons, Worksheets, Homework, Answer Keys & More!
Highlights
Description
POLYNOMIALS WHOLE UNIT for class 10 and 11! From an introduction to the polynomials unit [vocabulary words such as monomial, binomial, trinomial, term, degree, leading coefficient, divisor, quotient, dividend, etc.], then progresses deeper into the polynomials unit for how to calculate multiplicity, roots/zeros, end behavior, and finally sketching graphs of polynomials with varying degree and multiplicity. Students also match polynomial equations and their corresponding graphs. How to factor the polynomial - ALL methods, and how to solve polynomial equations by factoring. Polynomial Long Division, Synthetic Division, Factor and Remainder Theorems. How to identify intervals of increasing/decreasing, and [extrema] local/absolute max and min values. What is the Rational Roots Theorem? When and why do we use it? Finding all possible rational roots, actual rational roots, and all zeros of a polynomial function. Lessons are unique because they review important foundational skills FIRST.
Well-thought-out examples that flow, and excellent polynomial worksheets that tie everything together. The lessons make important connections and notes that students often miss; for example, about the relationships between real/imaginary zeros and x-intercepts.
✫ THIS PACKAGE IS A TOTAL OF 297 QUALITY PAGES!
INCLUDED:
- Detailed step-by-step answer keys and notes to EVERYTHING
- 9 Full lessons [PDF & SmartBoard Versions]
- 15 Worksheets
- 5 Reference sheets
- 2 Tests with study guide
- Creative degree>2 word problems
SUGGESTED ORDER:
1) How to FACTOR THE POLYNOMIAL [all methods step-by-step]
2) Solve Polynomial Equations BY FACTORING (which methods do I use!?)
3) FREE: Worksheet - Solve Equations by Formula or Factoring & Graphing
4) Intro, End Behavior, Multiplicity, Roots, Graph
5) Intervals of Increasing/Decreasing and Extrema [Local/Absolute Max & Min]
6) LONG DIVISION, FACTOR/REMAINDER THEOREMS
7) SYNTHETIC Division, FACTOR/REMAINDER Theorems
9) Degree>2 Real-World Word Problems
10) Polynomials UNIT TESTS with Study Guide
DETAILED LIST OF TOPICS:
➤ Definition: The word "polynomial" is broken down to "poly" and "nomial" ... which means many terms.
➤ Vocabulary words with examples: monomial, binomial, trinomial, term, degree, leading coefficient, root/zero, multiplicity, end behavior.
➤ Introduces multiplicity of roots and provides examples of:
- Determining a root and its corresponding multiplicity by looking at a polynomial equation in factored form.
- Sketching the graph of a polynomial function in factored form using the rules of multiplicity.
- Identifying a possible equation in factored form by looking at a polynomial graph with various multiplicity.
➤ Reviews:
- Identifying the degree and leading coefficient.
- Converting the equation to factored form by using the following factoring skills: by grouping, difference of cubes formula, and difference of squares formula.
- Using factored-form to help in finding the x-intercepts.
- Using the quadratic formula to produce 2 complex roots.
- Sketching the graph of a function where one of the roots has a multiplicity of 2, and knowing that two of the roots will not be graphed since they are complex.
➤ Students learn that the degree of a polynomial (whether it's even or odd) as well as the sign of the leading coefficient tie into determining the End Behavior [using proper notation]. Students then use this information to sketch a polynomial graph.
➤ Students sketch the graphs of polynomials with varying degree, using all the pieces of degree, roots/zeros, multiplicity and end behavior to help them.
➤ Makes very important connections that many students miss, if not pointed out, such as: "If a cubic function has 1 real root and 2 imaginary roots, how many x-intercepts will the function have? Draw a general picture."
➤ Students continue making important connections by sketching the graphs of quadratic and cubic functions with different numbers of x-intercepts. For example, sketching the graph of a quadratic function with 1 x-intercept versus none.
➤ Factoring - all methods:
1) Factor using the Greatest Common Factor (GCF) method
2) Factor by Grouping (4 terms)
3) Factor Quadratics (3 terms) - Both cases where leading coefficient (a) is equal to 1, and not equal to 1
4) Use the Difference of Perfect Squares formula to factor
5) Use the Sum & Difference of Cubes formula to factor
6) Factor a polynomial completely
➤ Long & Synthetic Division of polynomials.
➤ Factor/Remainder Theorems: Students are asked to determine whether the binomial is a factor of the dividend (only when the remainder=0); and if it is, then students can convert the polynomial into factored form by multiplying the divisor and quotient together.
➤ What is the Rational Roots Theorem? When and why do we use it? Finding all possible rational roots, actual rational roots, and all zeros of a polynomial function. Introductory lesson which reviews important foundational skills FIRST, and learning to distinguish whether or not Rational Roots Theorem is necessary in solving a polynomial equation.
➤ Creating polynomial word problem that reviews:
1) Identifying the x and y intercept(s) from the polynomial graph; then asking students to think about what do these points mean in the context of this stock market problem?
2) Finding intervals of increase and decrease.
3) Identifying the extrema (local/absolute max. and min. values). It also asks students to think about which extrema does not make sense in this real-world scenario?
4) Finding the exact equation in factored-form
5) Using the exact equation to make a future prediction
➤ 2 tests:
- Rearranging terms in an expression to convert into Standard Form.
- Identifying the degree and number of terms in an expression.
- Simplifying expressions.
- Factoring expressions.
- Identifying possible & exact equations of a polynomial graph in factored form using x-intercepts and multiplicity.
- Identifying x-intercepts and multiplicity from an equation.
- Solving equations by factoring and using the quadratic formula in simplest radical form.
- Identifying key features of a graph:
- Lowest possible degree
- X-and-Y-intercepts
- Intervals of increasing/decreasing
- Extrema (absolute & local max/min values)
- Operations (add/subtract/multiply and dividing using Long & Synthetic division)
