Polynomials

Lesson 14

Objective

Write polynomial functions from solutions of that polynomial function.

Common Core Standards

Core Standards

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  • 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

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  • F.BF.A.1

Criteria for Success

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  1. Notice and describe the pattern of number of points minus 1 defining the largest degree a set of points can define.
  2. Identify systems of equations as a viable process for determining the coefficients for the polynomial functions. 
  3. Describe that the number of points presented does not automatically determine the degree of the polynomial function.

Tips for Teachers

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In terms of pacing, this lesson can be taught over two days.

Anchor Problems

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Problem 1

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?

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 1 > Topic B > Lesson 20Example 1

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..

Problem 2

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? 

Guiding Questions

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Problem 3

What is the polynomial $$P$$ such that $$P(-1)=10$$, $$P(2)=1$$, and $$P(0)=3$$

Guiding Questions

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Problem Set

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

Target Task

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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.

References