Curriculum / Math / 11th Grade / Unit 3: Polynomials / Lesson 14
Math
Unit 3
11th Grade
Lesson 14 of 14
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Write polynomial functions from solutions of that polynomial function.
The core standards covered in this lesson
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.
The foundational standards covered in this lesson
F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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In terms of pacing, this lesson can be taught over two days.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Sarah is given two points that lie on a line.
$${f(8)=0 }$$
$${f(-2)=5}$$
She first writes the slope intercept form of the equation:
$${y=mx+b}$$
Then, she substitutes one point into the equation:
$${0=(8)m+b}$$
She then substitutes the other point into the slope intercept form:
$${5=(-2)m+b }$$
What are Sarah’s next steps in finding the equation of this line?
Algebra II > Module 1 > Topic B > Lesson 20 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
The general form for a polynomial function is:
$${{a_n}x^n+a_{n-1}x^{n-1}+a_1x+a_0 ...}$$
where $$n$$ is the degree of the polynomial, and $$a_{n}$$ is the leading coefficient.
So, for example: a 4th-degree polynomial has a general form of:
$${ax^4+bx^3+cx^2+dx+e}$$
What is the largest degree you could uniquely define if you were given three points?
What is the polynomial $$P$$ such that $$P(-1)=10$$, $$P(2)=1$$, and $$P(0)=3$$?
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Use the remainder theorem to find a quadratic polynomial $$P$$ so that $$P(1)=5$$, $$P(2)=12$$, and $$P(3)=25$$. Give your answer in standard form.
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.
Lesson 13
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.Â
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.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.
Identify the solution(s) to systems of polynomial functions.
A.REI.D.11
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