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Algebra I. Chapter 8/9 Notes Part II 8-5, 8-6, 8-7, 9-2, 9-3. Section 8-5: Greatest Common Factor, Day 1. Factors – Factoring – Standard Form Factored Form. Section 8-5: Greatest Common Factor, Day 1.

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algebra i

Algebra I

Chapter 8/9 Notes

Part II

8-5, 8-6, 8-7, 9-2, 9-3

section 8 5 greatest common factor day 1
Section 8-5: Greatest Common Factor, Day 1

Factors –

Factoring –

Standard Form Factored Form

section 8 5 greatest common factor day 11
Section 8-5: Greatest Common Factor, Day 1

Factors – the numbers, variables, or expressions that when multiplied together produce the original polynomial

Factoring – The process of finding the factors of a polynomial

Standard Form Factored Form

section 8 5 gcf day 1
Section 8-5: GCF, Day 1

Greatest Common Factor (GCF): The largest factor in a polynomial. Factor this out FIRST in every situation

Ex ) Factor out the GCF

1) 2)

3) 4) 15w – 3v

section 8 5 grouping day 2
Section 8-5: Grouping, Day 2

Factoring by Grouping

1) Group 2 terms together and factor out GCF

2) Group remaining 2 terms and factor out GCF

3) Put the GCFs in a binomial together

4) Put the common binomial next to the GCF binomial

Ex) 4qr + 8r + 3q + 6

section 8 5 grouping day 21
Section 8-5: Grouping, Day 2

Factor the following by grouping

1) rn + 5n – r – 5 2) 3np + 15p – 4n – 20

section 8 5 grouping day 22
Section 8-5: Grouping, Day 2

Factor by grouping with additive inverses.

1) 2mk – 12m + 42 – 7k

2) c – 2cd + 8d – 4

section 8 5 zero product property day 3
Section 8-5: Zero Product Property, Day 3

What is the point of factoring?

It is a method for solving non-linear equations (quadratics, cubics, quartics,…etc.)

Zero Product Property – If the product of 2 factors is zero, then at least one of the factors MUST equal zero.

Using ZPP:

1) Set equation equal to __________.

2) Factor the non-zero side

3) Set each __________

equal to ___________ and

solve for the variable

section 8 5 zero product property day 31
Section 8-5: Zero Product Property, Day 3

Solve the equations using the ZPP

  • (x – 2)(x + 3) = 0 2) (2d + 6)(3d – 15) = 0

3) 4)

section 8 6 factoring quadratics day 1
Section 8-6: Factoring Quadratics, Day 1

Factoring quadratics in the form:

Where a = 1, factors into 2 binomials:

(x + m)(x + n)

m + n = b the middle number in the trinomial

m x n = c the last number in the trinomial

Ex)  (x + 3)(x + 4)

section 8 6 factoring quadratics day 11
Section 8-6: Factoring Quadratics, Day 1

Factor the following trinomials

1) 2)

section 8 6 factoring quadratics day 12
Section 8-6: Factoring Quadratics, Day 1

Sign Rules:

 ( + )( + )

 ( - )( - )

 ( + )( - )

*If b is negative, the – goes with the bigger number

*If b is positive, the – goes with the smaller number

section 8 6 factoring quadratics day 13
Section 8-6: Factoring Quadratics, Day 1

Factor the following trinomials

1) 2)

3) 4)

section 8 6 solving quadratics by factoring day 2
Section 8-6: Solving Quadratics by Factoring, Day 2

Solve by factoring and using ZPP.

1) 2)

3) 4)

section 8 6 solving quadratics by factoring day 21
Section 8-6: Solving Quadratics by Factoring, Day 2

Word Problem: The width of a soccer field is 45 yards shorter than the length. The area is 9000 square yards. Find the actual length and width of the field.

section 8 7 the first last method when a does not 1 day 1
Section 8-7: The First/Last Method, when a does not = 1, Day 1

First/Last Steps:

1) Set up F, write factors of the first number (a)

2) Set up L, write factors of the last number (c)

3) Cross multiply. Can the products add/sub to get the middle number (b)? If not, try new numbers for F and L

Ex)

section 8 7 the first last method when a does not 1 day 3
Section 8-7: The First/Last Method, when a does not = 1, Day 3

Factoring using First/Last when c is negative.

1) 2)

section 8 7 factoring completely day 2
Section 8-7: Factoring Completely, Day 2

You must factor out a GCF FIRST! Then factor the remaining trinomial into 2 binomials.

1) 2)

section 8 7 solving by factoring day 21
Section 8-7: Solving by Factoring, Day 2

Lastly…Not all quadratics are factorable. These are called PRIME. It does not mean they don’t have a solution, it just means they cannot be factored.

Ex)

section 9 2 solving quadratics by graphing
Section 9-2: Solving Quadratics by Graphing

Solutions of a Quadratic on a graph:

section 9 2 solving quadratics by graphing1
Section 9-2: Solving Quadratics by Graphing

Solve the quadratics by graphing. Estimate the solutions.

Ex)

section 9 2 solving quadratics by graphing2
Section 9-2: Solving Quadratics by Graphing

Solve the quadratics by graphing. Estimate the solutions.

Ex)

section 9 2 solving quadratics by graphing3
Section 9-2: Solving Quadratics by Graphing

Solve the quadratics by graphing. Estimate the solutions.

Ex)

section 9 3 transformations of quadratic functions day 1
Section 9-3: Transformations of Quadratic Functions, Day 1

Transformation – Changes the position or size of a figure on a coordinate plane

Translation – moves a figure up, down, left, or right, when a constant k is added or subtracted from the parent function

section 9 3 transformations of quadratic functions day 12
Section 9-3: Transformations of Quadratic Functions, Day 1

Describe how the graph of each function is related to the graph of . First graph the parent function, then graph the given function.

a) b)

section 9 3 transformations of quadratic functions day 14
Section 9-3: Transformations of Quadratic Functions, Day 1

Describe how the graph of each function is related to the graph of . First graph the parent function, then graph the given function.

a) b)

section 9 3 transformations of quadratic functions day 15
Section 9-3: Transformations of Quadratic Functions, Day 1

Describe how the graph of each function is related to the graph of . First graph the parent function, then graph the given function.

a) b)

section 9 3 transformations of quadratic functions day 21
Section 9-3: Transformations of Quadratic Functions, Day 2

Describe how the graph of each function is related to the graph of . First graph the parent function, then graph the given function.

a) b)

section 9 3 transformations of quadratic functions day 23
Section 9-3: Transformations of Quadratic Functions, Day 2

Describe how the graph of each function is related to the graph of . First graph the parent function, then graph the given function.

a) b)