Lesson 6 | Polynomials | 11th Grade Mathematics

Objective

Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

F.BF.A.1.B — Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Building Functions

F.BF.A.1.B — Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Foundational Standards

The foundational standards covered in this lesson

A.REI.A.1

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.B.3

Reasoning with Equations and Inequalities

A.REI.B.3 — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Add like terms in polynomials.
Distribute the subtraction sign across polynomials and subtract terms accurately.
Predict the end behavior of sum or difference based on the leading coefficients of the polynomials.
Determine when a polynomial difference will result in a lesser-degree polynomial.
Use [VARS] "Y-VARS" "1. Function" to input Y1, Y2, Y3, etc., to graph the sum and difference of two functions
Graph multiple functions in [Y=] and [GRAPH]. "Turn on" and "turn off" functions in [Y=].

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

If $${f(x)= x^3-3x^2+2 }$$

and

$${g(x)=-4(x+2)^2-5}$$

What is $${ f(x)+g(x)}$$?
What is $${ f(x)-g(x)}$$?
What is $${g(x)-f(x)}$$?

Guiding Questions

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Problem 2

Is the following statement always, sometimes, or never true?
“The difference between two polynomials will be the same degree as the highest-degree polynomial in the difference.”

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Find $${h(x)-j(x)}$$.

$${h(x)=5x^4-3x^2+4}$$

$${j(x)=2x^3+5x^4+4}$$

What is the end behavior of the initial polynomials?
What is the end behavior of the resultant polynomial?
In what cases will a sum or difference of two polynomials with the same end behavior result in a polynomial with different end behavior?

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students need to write the original polynomials that result in a particular end behavior.
Include problems where there are missing coefficients or exponents that when added or subtracted result in a particular end behavior.
Include problems that confirm that polynomial addition is commutative but polynomial subtraction is not—following from properties of rational numbers.

EngageNY Mathematics Algebra I > Module 1 > Topic B > Lesson 8

Lesson 5

Lesson 7

Lesson Map

Topic A: Polynomial Features and Graphs

Classify polynomials through identification of degree and leading coefficient. Graph a polynomial function from a table of values; prove degree using successive differences.

A.APR.A.1 F.IF.C.7.C F.IF.C.9

Identify features of polynomial functions including end behavior, intervals where the function is positive or negative, and domain and range of function.

F.BF.B.3 F.IF.B.4 F.IF.B.6 F.LE.A.3

Match and compare equations and graphs of polynomials, and identify transformations.

F.BF.B.3 F.IF.B.4 F.IF.C.7.C

Sketch polynomial functions using sign charts and analysis of the factored form of the polynomial function.

A.APR.B.3

Topic B: Operations with Polynomials

Multiply polynomials, identifying factors, degrees, and number of real roots. Identify features of a polynomial in standard form.

A.APR.A.1 F.BF.B.3

Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.

A.APR.A.1 F.BF.A.1.B

Divide polynomials by binomials to determine linear factors.

A.APR.A.1 A.APR.B.2 A.APR.B.3

Determine if a binomial is a factor of a polynomial using the remainder theorem.

A.APR.B.2

Identify and factor with difference of two squares in quadratic and quartic polynomials. Describe identity of difference of two squares. Describe the zeros that represent the resultant factors.

A.APR.B.3 A.SSE.A.2

Factor difference and sums of cubes. Explain sums and differences of cubes as polynomial identities.

A.APR.C.4

Factor polynomials by grouping in quartic, cubic, and quadratic functions.

F.IF.C.8.A

Topic C: Polynomial Extensions

Use polynomial identities to determine Pythagorean triples.

A.APR.C.4

Identify the solution(s) to systems of polynomial functions.

A.REI.D.11

2 days

Write polynomial functions from solutions of that polynomial function.

A.APR.B.3

Polynomials

Objective

Common Core Standards

Core Standards

Arithmetic with Polynomials and Rational Expressions

Building Functions

Foundational Standards

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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