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Algebra II – Polynomial Function

This document provides study materials related to Algebra II – Polynomial Function. It may include explanations, summarized notes, examples, or practice questions designed to help students understand key concepts and review important topics covered in their coursework.

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Study GuideAlgebra II–Polynomial Function1. Polynomial FunctionWhat Is a Polynomial Function?Apolynomial functionis a type of mathematical expression made up of variables and numberscombined usingaddition, subtraction, and multiplication.The general form of a polynomial function looks like this:Understanding the Parts•(x)is the variable.•(a0, a1, a2, … ,an)are numbers calledcoefficients.•(n)is awhole number(0, 1, 2, 3, …).•The highest power of (x) determines thedegreeof the polynomial.Polynomial functions are evaluated bysubstituting a number for the variable (x)and thensimplifying the expression.For example, if we replace (x) with4in (P(x)), we write it as:This means“find the value of the polynomial when (x = 4)”.Example 1:Evaluating a PolynomialSuppose we have the polynomial:

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Study GuideWe want to find(P(-4)).Step 1:Substitute (x =-4)Step 2:Calculate the powersStep 3:MultiplyStep 4:Add and subtractThis means when (x =-4), the value of the polynomial is-241.Example 2:Finding (f(x + h))Now consider the function:We want to find(f(x + h)).This means wereplace every (x) with (x + h).Step 1:Substitute (x + h)

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Study GuideStep 2:Expand the squareSo,Step 3:MultiplyFinal ExpressionThis is the expanded form of(f(x + h)).2. Quiz: Polynomial Function1. QuestionIf (P(y) = 2y4-3y3+ 4y2-7), find (P(3)).Answer Choices•110•81•29Correct Answer:110

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Study GuideWhy This Is CorrectTo find (P(3)), substitute (y = 3) into the polynomial.Now calculate each term:Therefore, the correct answer is110.2. QuestionIf (Q(z) = 3z3+ 4z2-7z + 8), find (Q(-5)).Answer Choices•232•518•-232Correct Answer:-232Why This Is CorrectSubstitute (z =-5) into the polynomial.Calculate each part:

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Study GuideSo the correct answer is-232.3. QuestionIf (f(x) = x3+ 6x + 9), find (f(x-2)).Answer Choices•(x3+ 6x2+ 8x + 9)•(x3-6x2+ 18x-11)•(x3-6x2-8x + 9)Correct Answer:(x3-6x2+ 18x-11)Why This Is CorrectSubstitute (x-2) into the function.First expand the cube:Nowsubstitute:Combine like terms:

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Study GuideThus the correct answer is(x3-6x2+ 18x-11).4. QuestionIf (f(x) = x2-4x-3), find (f(x + 3)).Answer Choices•(x2-2x-6)•(x2+ 2x-6)•(x2+ 2x + 6)Correct Answer:(x2+ 2x-6)Why This IsCorrectSubstitute (x + 3) into the function.Expand:Combine like terms:So the correct answer is(x2+ 2x-6).

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Study Guide5. QuestionIf (g(x) = 2x3+ 4x-5), find (g(x-m)).Answer Choices•(2x3-m2x2+ 4x-4m-2m3-5)•(2x3+ 12m2x2+ 4x-4m-2m3-5)•(2x3-6mx2+ 6m2x + 4x-4m-2m3-5)Correct Answer:(2x3-6mx2+ 6m2x + 4x-4m-2m3-5)Why This Is CorrectSubstitute (x-m) into the function.First expand the cube:Multiply by 2:Now include theremaining terms:Final expression:Thus the correct answer is(2x3-6mx2+ 6m2x + 4x-4m-2m3-5).

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Study Guide3. Remainder TheoremWhat Is the Remainder Theorem?TheRemainder Theoremhelps us quickly find the remainder when a polynomial is divided by alinear expression.It states:If a polynomial (P(x)) is divided by ((x-r)), then theremainder is equal to (P(r)).In simple termsInstead of performing long division, you can:1.Replace (x) with (r) in the polynomial.2.Calculate the value.3.That value is theremainder.So,This makes calculationsmuch faster and easier.Example 1:Finding (P(-3))Given:We want to find:

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Study GuideThere aretwo possible methodsto solve this.Method 1: Direct SubstitutionSimply replace (x) with−3.Step 1:Evaluate the powersStep 2:Substitute the valuesStep 3:MultiplyStep 4:SimplifyFinal Answer:Method 2: Using Synthetic Division

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Study GuideAnother way to find the remainder is bysynthetic division.Since we want (P(-3)), we divide the polynomial by:which simplifies to:Using synthetic division with−3, we calculate the remainder.After completing the synthetic division steps, theremainder is −1610.This confirms our result fromMethod 1.Therefore,Example 2:Finding the RemainderFind the remainder when:is divided by:Method 1:Apply the Remainder TheoremAccording to the theorem,Step 1:Substitute (x = 4)
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Document Details

University
Texas A&M University
Course
Algebra II
Subject
Mathematics

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