Curriculum / Math / 11th Grade / Unit 3: Polynomials / Lesson 10
Math
Unit 3
11th Grade
Lesson 10 of 14
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Factor difference and sums of cubes. Explain sums and differences of cubes as polynomial identities.
The core standards covered in this lesson
A.APR.C.4 — Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)2 = (x² — y²)² + (2xy)² can be used to generate Pythagorean triples.
The foundational standards covered in this lesson
A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
The diagram below shows a cube of dimensions $$B$$ subtracted from a cube of dimensions $$A$$. The total volume represents $$ A^3-B^3$$.
The diagram below shows the volume of $$ A^3-B^3$$ subdivided into smaller volumes. Find the formula for the difference of two cubes using the smaller volumes shown.
“Difference of Cubes” by Alex Toole by Alex Toole is made available by YouTube under the Standard YouTube License. Accessed Sept. 26, 2017, 3:13 p.m..
Determine if the following are factors of the cubic polynomials shown:
Is $${(a-b)}$$ a factor of $${a^3-b^3}$$?
Is $${(a+b)}$$ a factor of $${a^3+b^3}$$?
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Factor the following expressions and verify that the factored expression is equivalent to the original.
$${16x^6+2y^3}$$
$${16x^6-2y^3}$$
How are the two factored expressions different?
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.
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Factor polynomials by grouping in quartic, cubic, and quadratic functions.
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.
Standards
A.APR.A.1F.IF.C.7.CF.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.3F.IF.B.4F.IF.B.6F.LE.A.3
Match and compare equations and graphs of polynomials, and identify transformations.
F.BF.B.3F.IF.B.4F.IF.C.7.C
Sketch polynomial functions using sign charts and analysis of the factored form of the polynomial function.
A.APR.B.3
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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.1F.BF.B.3
Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.
A.APR.A.1F.BF.A.1.B
Divide polynomials by binomials to determine linear factors.
A.APR.A.1A.APR.B.2A.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.3A.SSE.A.2
A.APR.C.4
F.IF.C.8.A
Topic C: Polynomial Extensions
Use polynomial identities to determine Pythagorean triples.
Identify the solution(s) to systems of polynomial functions.
A.REI.D.11
2 days
Write polynomial functions from solutions of that polynomial function.
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