Algebra I Unit 2

Unit 2:

Seeing Structure In Expressions

Section 1.

Interpret Expressions and Formulas

Vocabulary

‐Polynomial Expression: the sum of a finite number of
terms

‐Term: product of a real number and a variable

‐Coefficient: A real number attached to a variable

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Vocabulary Continued

‐Rational Expressions: the ratio of two polynomial
expressions

‐Numerator: The top portion of a fraction

‐Denominator: The bottom portion of a fraction

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Examples

Example 1. Mark is moving. He is packing a box, which is in
the shape of a rectangular prism. Mark notices that the base
of the box is in the shape of a square. The volume of the box,
V(s), is given by the following function, where s is the side
lengths, in inches, of the base of the box. Describe the factor
(s+14).

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Examples

Example 2. Wilma is conducting a bake sale at her school, where
she is selling cookies. In order to conduct the bake sale, she spends
$12.50 for supplies. Wilma is charging $1.25 per cookie. The
average profit per cookie, C(x), is given by the rational function
below, where x is the number of cookies that Wilma sells. What
does the numerator of C(x) represent?

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Examples

Example 3. A class visited the aquarium for a field trip. There were x students and y
teachers who visited the aquarium. The tickets to the aquarium cost s dollars for the
students. The expression below represents the total amount spent on teacher's tickets to
the aquarium. Which statement is the best interpretation of the expression in everyday
language?

y(s+5)

a.The number of teachers who attended the field trip was 5 more than the number of
students who attended.

b.The number of students who attended the field trip was 5 more than the number of
teachers who attended.

c.The price of a teachers ticket was 5 more than the price of a students ticket.

d.The price of a students ticket was 5 more than the price of a teachers ticket.

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Examples

Example 4. The total amount Georgia paid for her lunch, including tax
and tip, is represented by the expression below. Which statement
best describes 0.0825x?

a.

It’s the amount of her lunch entree

b.

It’s the amount of the tip

c.

It’s the amount of the tax

d.

It’s the amount of her beverage

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Examples

Example 5. The area of a triangular pennant is
represented by the following equation, where b
represents the base in centimeters of the pennant. Write
a statement that best describes (4b+6).

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Section 2.1

Factoring Expressions and Equations

Common Factoring

Vocabulary

‐Greatest common factor: largest factor in both terms

‐Distributive property: a(b+c)= ab+ac

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Steps to Common Factoring

1.

Find the greatest common factor

2.

Pull out the GCF

3.

Check your work using the distributive property

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Example 1 and 2

1.

2x(3x)+2x(5)

2.

2x3-6x2

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Examples 3, 4, and 5

3. 12x2+18x

4. 10x2+25x+15

5. x4-8x3+x2

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Section 2.2

Factoring Expressions and Equations

Sum-Product Method

When to Use Sum-Product Method

When the factors of c add up to b

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Steps To Doing Sum-Product Method

1.

Find the factors of c

2.

Find which factors give you the sum of b

3.

Set up factors so they will distribute to giving you the
original expression

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Examples

1.

X2+2x-35

2.

X2+6x-55

3.

x2+18x+77

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Section 2.3

Factoring Expressions and Equations

Difference of Squares and Perfect Squares

When To use:

Difference of Squares: when the polynomial is the form
ax2-c and c is a perfect square

Perfect Squares: When the polynomial is in the form
ax2+bx+c or ax2-bx+c and is a perfect square with the
sum of b

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Steps For Difference of Squares

1.

Find the factors of the square

2.

Factor the trinomial (one positive, one negative)

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Examples 1 and 2

1.

X2-9

2.

x2-36

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Steps for Perfect Square Method

1.

Determine whether the first and last terms are
perfect squares

2.

Factor so that the factors are the same dependent
on whether b is positive or negative

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Examples 1, 2, and 3

1.

X2+10x+25

2.

X2-12x+36

3.

x2+14x+49

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Section 2.4

Factoring Expressions and Equations

Factor by Grouping

When to Use the Grouping Method:

When the polynomial is the form ax2+bx+c and a.1 or
the polynomial is the form ax3+bx2+cx+d

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Steps when the polynomial is the second power

1.

Find the product of a(c)

2.

Find the factors of a(c)

3.

Find the factors that combine to make b

4.

Rewrite the polynomial with the new terms taking
b’s place

5.

Isolate similar terms

6.

Factor out the GCF from the similar terms

7.

Combine new terms

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Steps when the polynomial is the third degree

1.

Group the first two and last two terms together

2.

Factor out the GCF from both groups

3.

Combine new terms

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Examples 1 and 2

1.

6x2+7x+2

2.

2x2+7x+3

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Examples 3 and 4

3. 14x3-10x2+21x-15

4.30k3+35k2+24k+28

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Section 3:

Factoring to Reveal Properties

Vocabulary

‐Quadratic Equations: equation in the form ax2+bx+c

‐Zeros/Roots: What x equals when y equals zero and
where the graph intersects the x axis

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Examples

1.

Solve for x. x2+9x+8=0

2.

Solve for x. x2+4x=21

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Completing the Square

Write an equivalent statement by completing the
square.

x2+16x-58

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Completing the Square

Find the minimum or maximum by completing the
square.

x2-8x-7

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