name: chapter 8: quadratic expressions and equations...

Name: Chapter 8: Quadratic Expressions and Equations

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Lesson 8-1: Adding and Subtracting Polynomials Date:

A is a monomial or the sum of monomials, each called a term of the polynomial.

A is the sum of two monomials.

A is the sum of three monomials.

Examples of polynomials:

monomial binomial trinomial

The degree of a monomial is the of the exponents of all its variable.

The degree of a polynomial is the degree of any term in the polynomial.

Degree Name Example

0

1

2

3

4

5

6 or more

Example 1: State whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial, or trinomial.

A. 6𝑥 − 4 B. 𝑥2 + 2𝑥𝑦 − 7

C. 14𝑑+19𝑒3

5𝑑4 D. 26𝑏2


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The form of a polynomial has the terms in order from greatest to least degree.

The coefficient of the first term is called the coefficient.

Example 2: Write each polynomial in standard form. Identify the leading coefficient.

A. 9𝑥2 + 3𝑥6 − 4𝑥 B. 12 + 5𝑦 + 6𝑥𝑦 + 8𝑥𝑦2

Example 3: Find each sum.

A. (7𝑦2 + 2𝑦 − 3) + (2 − 4𝑦 + 5𝑦2) B. (4𝑥2 − 2𝑥 + 7) + (3𝑥 − 7𝑥2 − 9)

Example 4: Find each difference.

A. (6𝑦2 + 8𝑦4 − 5𝑦) − (9𝑦4 − 7𝑦 + 2𝑦2)

B. (6𝑛2 + 11𝑛3 + 2𝑛) − (4𝑛 − 3 + 5𝑛2)


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Example 5: The total amount of toy sales T (in billions of dollars) consists of two groups: sales of video games V and sales of traditional toys R. In recent years, the sales of traditional toys and total sales could be modeled by the following equations, where n is the number of years since 2000.

𝑅 = 0.46𝑛3 − 1.9𝑛2 + 3𝑛 + 19

𝑇 = 0.45𝑛3 − 1.85𝑛2 + 4.4𝑛 + 22.6

A. Write an equation that represents the sales of video games V.

B. Use the equation to predict the amount of video game sales in the year 2009.


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Lesson 8-2: Multiplying a Polynomial by a Monomial Date:

Example 1: Simplify each expression.

A. 6𝑦(4𝑦2 − 9𝑦 − 7) B. 3𝑥(2𝑥2 + 3𝑥 + 5)

Example 2: Simplify each expression.

A. 3(2𝑡2 − 4𝑡 − 15) + 6𝑡(5𝑡 + 2) B. 5(4𝑦2 + 5𝑦 − 2) + 2𝑦(4𝑦 + 3)

Example 3: Admission to the Super Fun Amusement Park is $10. Once in the park, super rides are an additional $3 each and regular rides are an additional $2. Wyatt goes to the park and rides 15 rides, of which s of those 15 are super rides. Find the cost if Wyatt rode 9 super rides.

Example 4: Solve 𝑏(12 + 𝑏) − 7 = 2𝑏 + 𝑏(−4 + 𝑏)


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Lesson 8-3: Multiplying Polynomials Date:

A is an expression in one variable with a degree of 2.

Example 1 & 2: Find each product.

A. (𝑦 + 8)(𝑦 − 4)

B. (2𝑥 + 1)(𝑥 + 6)


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Example 3: A patio in the shape of the triangle shown is being built in Lauren’s backyard. The dimensions given are in feet. The area A of the triangle is one half the height h times the base b. Write an expression for the area of the patio.

Example 4: Find each product.

A. (3𝑎 + 4)(𝑎2 − 12𝑎 + 1)

B. (2𝑏2 + 7𝑏 + 9)(𝑏2 + 3𝑏 − 1)


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Lesson 8-4: Special Products Date:

Example 1:

A. (7𝑧 + 2)2 B. (3𝑥 + 4)2

Example 2:

A. (3𝑐 − 4)2 B. (2𝑚 − 3)2


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Example 3: Write an expression that represents the area of a square that has a side length of 3x + 12 units.

Example 4:

A. (9𝑑 + 4)(9𝑑 − 4) B. (3𝑦 − 2)(3𝑦 + 2)


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Lesson 8-5: Using the Distributive Property Date:

Example 1: Use the distributive property to factor each of the following expressions.

A. 15𝑥 + 25𝑥2 B. 12𝑥𝑦 + 24𝑥𝑦2 − 30𝑥^2𝑦4

Example 2: Factor each of the following.

A. 2𝑥𝑦 + 7𝑥 − 2𝑦 − 7 B. 4𝑥𝑦 + 3𝑦 − 20𝑥 − 15

Example 3: Factor 15𝑎 − 3𝑎𝑏 + 4𝑏 − 20


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Example 4: Solve each equation. Check the solution.

A. (𝑥 − 2)(4𝑥 − 1) = 0 B. 4𝑦 = 12𝑦2

Example 5: A football is kicked into the air. The height of the football can be modeled by the equation

ℎ = −16𝑥2 + 48𝑥, where ℎ is the height reached by the ball after 𝑥 seconds. Find the values of 𝑥 when

ℎ = 0.


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Lesson 8-6: Solving 𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 Date:

Example 1: Factor x2 + 7x + 12.

Example 2: Factor x2 – 12x + 27.


A. x2 + 3x – 18. B. x2 – x – 20

Example 4: Solve x2 + 2x = 15. Check your solution.


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Example 5: Marion wants to build a new art studio that has three times the area of her old studio by increasing the length and width by the same amount. What should be the dimensions of the new studio?


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Lesson 8-7: Solving 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 Date:


A. 5x2 + 27x + 10

B. 4x2 + 24x + 32


A. 24x2 – 22x + 3

B. 10x2 – 23x + 12


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Example 3: Factor 3x2 + 7x – 5, if possible.

Example 4: Mr. Nguyen’s science class built a model rocket for a competition. When they launched their rocket outside the classroom, the rocket cleared the top of a 60-foot high pole and then landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation h = –16t2 + vt + h0.


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Lesson 8-8: Difference of Squares Date:


A. m2 – 64 B. 16y2 – 81z2

C. 3b3 – 27b


A. y4 – 625 B. 256 – n4


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A. 9x5 – 36x B. 6x3 + 30x2 – 24x – 120

MULTIPLE CHOICE Example 4: In the equation 𝑞2 −4

25= 𝑦 which is a value of q when y = 0?

A. 2

25

B. 4

25

C. 0

D. −2

5


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Lesson 8-9: Perfect Squares Date:

Step 1: Is the first term a perfect square?

Step 2: Is the last term a perfect square?

Step 3: Is the middle term equal to 2𝑎𝑏?

Example 1: Determine whether each of the following are perfect square trinomials. If so, factor them.

A. 25𝑥2 − 30𝑥 + 9 B. 49𝑦2 + 42𝑦 + 36


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A. 6𝑥2 − 96 B. 16𝑦2 + 8𝑦 − 15

Example 2: Solve 4𝑥2 + 36𝑥 = −81.


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Example 4: Solve each of the following.

A. (𝑏 − 7)2 = 36 B. (𝑥 + 9)2 = 8

Example 5: A book falls from a shelf that is 5 feet above the floor. A model for the height h in feet of an object dropped from an initial height of h0 feet is h = –16t2 + h0 , where t is the time in seconds after the object is dropped. Use this model to determine approximately how long it took for the book to reach the ground.

name: chapter 8: quadratic expressions and equations...

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