Lesson 7 | Polynomials | 11th Grade Mathematics

Objective

Divide polynomials by binomials to determine linear factors.

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.
A.APR.B.2 — Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

Arithmetic with Polynomials and Rational Expressions

A.APR.B.2 — Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
A.APR.B.3 — Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Arithmetic with Polynomials and Rational Expressions

A.APR.B.3 — Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

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

Describe that just like with the operations of multiplication, addition, and subtraction, polynomial division follows the same principles as integer division.
Divide binomials into polynomials using long division.
Identify that when the remainder of this division is zero, the binomial is a factor of the polynomial.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In this lesson, students will focus on long division and not be introduced to synthetic division. For more information about reasoning for this, see the Algebra Progressions for the Common Core, page 9. Lane Walker also provides some additional reasoning on the topic. She addresses the topic again in this post.

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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

Find the product.

$${(x-3)(x^2-2x+3)}$$

Once we find the product, how would we work backwards to get both factors again?

Guiding Questions

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Problem 2

What is the other factor you would multiply $${x-2}$$ by to get the polynomial $${x^4-4x^2-5x+10}$$?

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

Problem 1

One factor of the polynomial $${2x^4+2x^3+3x+3}$$ is $${(x+1)}$$. What is the other factor?

Problem 2

How can you confirm that $${x+1}$$ is indeed a factor of $${2x^4+2x^3+3x+3}$$ through the long division?

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 are given the graph of the polynomial function (with an integer root), the equation, and they need to identify one linear factor from the root and then show this as a factor through long division.
Include problems where students are given the roots and they need to test one of the factors through long division.

EngageNY Mathematics Algebra II > Module 1 > Topic A > Lesson 3
Mathematics Vision Project: Secondary Mathematics Three Module 3: Polynomial Functions — Lesson 3.7, Problems 1-14

Lesson 6

Lesson 8

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

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

Foundational Standards

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Problem 1

Problem 2

Additional Practice

Lesson Map

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