9th,Per.3,Team+4

Chapter 8: Polynomials and Factoring Justin, Cody , Emily

Emily's Review and Summarization of Chapter 8 Section 1

Chapter 8-1 Adding and Subtracting Polynomials

EXAMPLE 1 : Degree of a Monomial. A monomial is an expression that is a number, a variable, or a product of a number and one or more variables. Each of the following is a monimial. 12 y -5x^2y c/3

The fraction c/3 is a monomial, but the expression c/x, or cx^-1, is NOT a monomial because the exponent of the variable is negative.

The degree of a monomial is the sum of the exponents of its variables. For a nonzero constant, the degree is 0. Zero has no degree.

Degree of a Monomial

a. 2/3x Degree: 1 2/3x = 2/3x^1. The exponent is 1.

7x^2y^3 Degree: 5 The exponents are 2 and 3. Their sum is 5.

-4 Degree: 0 The degree of a nonzero constant is 0.

Find the Degree of -5xy^2.

A Polynomial is the sum of one or more monomials.

3x^4 + 5x^2 - 7x + 1

Degree > 4 2 1 0

The polynomial show above is the standard form. Standard form of a polynomial means that the degrees of its monomial terms decrease from left to right. The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. The degree of 3x^4 + 5x^2 - 7x + 1 is 4.

EXAMPLE 2 : Adding Polynomials. Simplify (4x^2 + 6x + 7) + (2x^2 - 9x +1)

Step 1: Line up like terms. Then add.

(4 x^2 + 6 x +7) + (2 x^2 + 9 x +1)

6x^2 - 3 x + 8

Although this method is easy, there is annother way you could have solved this problem.

Add Horizontally. Group like term. Then add.

(4 x^2 + 6 x + 7) + (2 x^2 - 9 x +1) = (4 x^2 + 2 x^2 ) + (6 x - 9 x ) + (7 + 1) = 6 x^2 - 3 x + 8

EXAMPLE 3 :  Subtracting Polynomials .     Simplify (2x^3 + 5x^2 - 3x) - (x^3 - 8x^2 + 11)   Method 1. Subtract Vertically.  <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> 2x^3 + 5x^2 - 3x Line up Like Terms. <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> (x^3 - 8x^2 + 11) <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> - <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> 2x^3 + 5x^2 - 3x <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> <span style="color: rgb(112, 32, 141);">- <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> x^3 <span style="color: rgb(112, 32, 141);">+ <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> 8x^2 <span style="color: rgb(112, 32, 141);">- <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> 11 Then add the opposite or each term in the <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> polynomial being subtracted. <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> --- <span style="color: rgb(112, 32, 141);"> <span style="letter-spacing: 0px; color: rgb(112, 32, 141);"> x^3 + 13x^2 - 3x - 11

Justin's Review and Summarization of 8-2

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Codys Review and Summarization of Chapter 8.3

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Cody McDougald Chapter 8.3

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<span style="color: rgb(0, 128, 0);">Chapter 8.3 Multiplying Binomials
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<span style="color: rgb(0, 128, 0);">Example 1. Using the distributive property!
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<span style="color: rgb(0, 128, 0);">Simplify (2x+3)(x+4)
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<span style="color: rgb(0, 128, 0);">(2x+3)(9x+4)=2x(x+4)+3(x+4)--Distribute x+4
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<span style="color: rgb(0, 128, 0);">= 2x^2+8x+3x+12 Now distribute 2x and 3 Simplify.
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<span style="color: rgb(0, 128, 0);">Example 2. Multiplying using FOIL.
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<span style="color: rgb(0, 128, 0);">Simplify (3x-5)(2x+7)
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<span style="color: rgb(0, 128, 0);">(3x-5)(2x+7) = First =(3x)(2x)
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<span style="color: rgb(0, 128, 0);">Outer=(3x)(7) Inner=(5)(2x) Last=(5)(7)
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<span style="color: rgb(0, 128, 0);">solve=6x^2 + 21x -10x - 35 =6x^2 + 11x - 35
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<span style="color: rgb(0, 128, 0);">The product is - 6x^2 + 11x - 35
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<span style="color: rgb(0, 128, 0);">Example 3. Multiplication of polynomials.
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<span style="color: rgb(0, 128, 0);">area of the outer rectangle = (3x+1)(2x+5)
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<span style="color: rgb(0, 128, 0);">area of hole= x(x+2)
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<span style="color: rgb(0, 128, 0);">area of shade region
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<span style="color: rgb(0, 128, 0);">= area of outer rectangle - area of hole = (3x+1)(2x+5)- x(x+2) subtract
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<span style="color: rgb(0, 128, 0);">= 6x^2+15x+2x+5-x^2-2x (use the foil to simplify)
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<span style="color: rgb(0, 128, 0);">(3x+1)(2x+5)
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<span style="color: rgb(0, 128, 0);">=6x^2-x^2+15x+2x-2x+5= Group like terms
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