1

Unit 8 Notes: Solving Quadratics by Factoring Alg 1

Name_____________________________ Period _______

NOTE: You should be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit, or else Homework Club will be assigned! (Students with 100% Homework completion at the end of the semester will be rewarded

with a pizza party and 2% grade increase.) HW reminders:

If you cannot solve a problem, get help before the assignment is due. Need help? Try DRHSalgebra1.weebly.com. Extra Help? Visit www.mathguy.us or www.khanacademy.com.

8.1 Introduction to Polynomials

Monomial

Binomial

Trinomial Polynomial

Note: All the exponents must be

____________________________ numbers!

Day Date Assignment (Due the next class meeting) Tuesday

Wednesday

3/4/14 (A)

3/5/14 (B)

8.1 Worksheet

Intro to Polynomials

Thursday

Friday

3/6/14 (A)

3/7/14 (B)

8.2 Worksheet

Multiplying Binomials

Monday

Tuesday

3/10/14 (A)

3/11/14 (B)

8.3 Worksheet

Factoring and Solving by GCF and Grouping

Wednesday

Thursday

3/12/14 (A)

3/13/14 (B)

8.4 Worksheet

Factoring and Solving Trinomials

Friday

Monday

3/14/14 (A)

3/17/14 (B)

8.5 Worksheet

Practice Factoring Day

Tuesday

Wednesday

3/18/14 (A)

3/19/14 (B)

8.6 Worksheet

Factoring and Solving Trinomials with a leading coefficient

Thursday

Friday

3/20/14 (A)

3/21/14 (B)

8.7 Worksheet

More Factoring and Solving

Monday

Tuesday

3/24/14 (A)

3/25/14 (B) Unit 8 Practice Test

Wednesday

Thursday

3/26/14 (A)

3/27/14 (B) Unit 8 Test

2

Degree of a polynomial

Leading Coefficient

Descending order

2nd Differences. Tell whether each is a linear or quadratic function x 1 2 3 4

y 9 12 17 24

Adding polynomials

Example 1: (4x3 + x2 – 5) + (7x + x3 - 3x2).

Example 2: Find the sum: (x2 + x + 8) + (x2 – x – l)

Subtracting polynomials Example 3: Find the difference: (4z2 - 3) – (-2z2 + 5z - l).

Example 4: Find the difference of (3x2 + 6x – 4) – (x2 – x –7).

x 1 2 3 4

y 13 9 5 1

Remember to

multiply each

term in the

polynomial by – 1

when you write

the subtraction

as addition.

3

Objective #1: Can you add and subtract polynomials?

a) (3m3 + 2m + 1) + (4m2 – 3m + 1) b) (–4c + c3 + 8) + (c2 – 5c – 3)

c) (3x2 + 5) – (x2 + 2) + (– 3m + 1) d) (–3z + 6) – (4z2 – 7z – 8)

2 good ways to multiply polynomials:

1. Use distribution properties

2. Use box method Step 1: Decide the dimensions of the box your will draw. ______ by # of terms

Step 2: Draw the box and label each __________________.

Step 3: Multiply the terms for each part of your box.

Step 4: Combine like terms and write your answer in descending order.

Example 5: Find the product 3x3(2x3 – x2 – 7x – 3).

Example 6: Multiply −𝑥2(𝑥 − 6)

4

Objective #2: Can you multiply monomials and polynomials?

a) 4𝑥(𝑥 − 3) b) 4𝑥2(3𝑥4 − 2𝑥 + 7) c) −9𝑥(𝑥2 + 5)

8.2: Multiplying Polynomials Warm-Up

Simplify the following:

1) 2𝑥3 ∙ 3𝑥5 ∙ 2𝑥 2) −3(2𝑥 − 9) 3) (2𝑠3 − 𝑠2 + 1) − (3𝑠2 − 𝑠 + 4)

Multiplying polynomials (Box Method) Step 1: Decide the dimensions of the box your will draw. ______ by ______

Step 2: Draw the box and label each __________________.

Step 3: Multiply the terms for each part of your box.

Step 4: Combine like terms and write your answer in descending order.

Example 1:

a. (2𝑎– 5)(𝑎2– 6𝑎– 3) b. (3b + 5)(5b – 6)

5

Alternative to the box method: Multiplying binomials using the FOIL pattern

F O I L

Then combine like terms and write in descending order.

Example 3: Find h(x) = f(x) ∙ g(x) if Example 4: Use FOIL to multiply (x + 3)(3x + 8)

f(x) = (2x + 7) and g(x) = (x – 9).

Objective #3: Can you multiply polynomials?

a) (x – 7)(3x + 4) b) (a – 5)(a + 3)

c) Find h(x) = f(x) ∙ g(x) if f(x) = (3x + 2) and g(x) = (3x – 2)

d) (m + 7)(m – 3) + (m – 4)(m + 5)

6

Reflect #1: In FOIL, which of the products combine to form the x-term? Which products combine

to form the constant term? What about the x²-term?

x-term:

constant term:

x²-term:

Finding Special Products of Polynomials SQUARE OF A BINOMIAL *What would you expect the product of (𝑥 + 3)2 to be?

Foil it out!

Was your prediction correct?

*Predict the product of (𝑥 − 5)2

Foil it out!

Was your prediction correct?

Example 5: Find each square of a binomial.

a) (x + 4)2 b) (3x – 2)2

7

Objective #4: Can you square binomials?

a) (x + 5)2 b) (m – 8)2 c) (3c – 2)2

SUM AND DIFFERENCE PATTERN (also called multiplying conjugates) Example 6: Find each product. Do you see a pattern?

a) (x + 4)(x – 4) b) (2a – 7b) (2a + 7b)

The Sum and Difference Pattern described using algebra:

(a + b)(a – b) = ____2 – ____2

Objective #5: Can you use the Sum and Difference Pattern to find the product

of two binomials?

a) (x + 5)(𝑥 − 5) b) (m – 8n)(𝑚 + 8𝑛) c) (3c + 2)(3𝑐 − 2)

8

8.3 Guided Notes: Factor and Solve using Greatest Common Factor (GCF) Warm-Up

Simplify the following:

1) 2𝑥(𝑥 − 5𝑥3) 2) 2√5(√20 − 4) 3) (x + 3)(x – 4)

FACTORING out a greatest common factor (Box Method). Step 1: Write the terms _______________ the boxes.

Step 2: Pull out the greatest common factor to the ___________________.

Step 3: Write the other product at the tops of the boxes.

Step 4: Write your answer as a monomial times a binomial.

Example 1: Factor out the greatest common factor (GCF).

a) 5x + 20 b) 8x – 4x2

c) 16x + 40y + 8 d) 6x2 – 30x3

Objective #6: Can you factor polynomials using the Greatest Common Factor?

a) 4m – 2 b) -5x2 – 10x c) 6y + 15 d) -9m3 + m2 – 2m

9

Factoring by grouping (the Box Method) Step 1: FACTOR OUT GCF!!!

Step 2: Draw a 2x2 box.

Step 3: Write your terms in descending order.

Step 4: Put one ____________ in each section of the box.

Step 5: Pull out a Greatest Common Factor (to the _____________________.)

Step 6: Write your answer as a product of two binomials.

Example 2: Factor each expression.

a) 6𝑦3 + 3𝑦2 + 12𝑦 + 6 b) −3𝑥3 + 6𝑥2 − 15𝑥 + 30

c) −𝑥4 + 12𝑥 − 3𝑥2 + 4𝑥3 d) 6𝑎3 − 3𝑎2 + 8𝑎 − 4

Objective #7: Can you use grouping to factor polynomials?

a) 10𝑥3 + 40𝑥2 − 20𝑥 − 80 b) −𝑥3 + 5𝑥2 − 2𝑥 + 10

c) 𝑥3– 4𝑥2– 6𝑥 + 24 d) 3𝑥4 − 4𝑥3 + 5𝑥2 − 20𝑥

10

Reflect #3: Explain what we are doing when we factor a polynomial.

Zero-Product Property

Let a and b be real numbers. If ab = 0, then _____= 0 or _____ = 0.

Roots of an equation:

Also called…

Example 3: Solve each equation.

a) (x – 5)(x + 4) = 0 b) 2a(6a + 1) = 0

Example 4: Solve 2𝑥2 − 32𝑥 = 0 by factoring. Sketch a graph that includes the x-intercepts.

11

Example 5: Solve −2𝑥2 − 8𝑥 = 0 by factoring. Sketch a graph that includes the x-intercepts.

Objective #8: Can you solve equations by factoring?

a) (𝑥 + 2)(𝑥 − 8) = 0 b) 4𝑥2 + 24𝑥 = 0 c) −8𝑝2 − 24𝑝 = 0

8.4: Factoring and Solving Trinomials and a Difference of Perfect Squares

Warm-Up

1) What is the simplified form of (𝑥 − 6)2 ?

A. 𝑥2 − 12𝑥 − 36 B. 𝑥2 − 12𝑥 + 36

C. 𝑥2 − 36 D. 𝑥2 + 36

2) What is the product of (𝑥 + 3)(𝑥2 + 2𝑥 − 4) ?

12

Work in groups to FOIL the following:

(𝑥 + 5)(𝑥 − 3) (𝑥 + 2)(𝑥 + 8) (𝑥 + 3)(𝑥 + 3)

(𝑥 − 5)(𝑥 − 4) (𝑥 − 3)(𝑥 − 10) (𝑥 − 5)(𝑥 + 8)

Reflect #4: What are the patterns? How can you short-cut this?

Factoring is the opposite of FOILing… Example 1: Factor x2 + 6x + 5

Example 2: Factor each expression.

a) x2 + 10x + 16 b) a2 – 5a + 6 c) 𝑥2 + 7𝑥 − 30

d) 𝑏2 + 10𝑥 + 9 e) 𝑦2 − 𝑦 − 6 f) 𝑥2 + 2𝑥 + 1

g) 3y2 + 9y – 30 h) −𝑥2 + 4𝑥 + 12 i) 𝑤3 − 7𝑤2 − 30𝑤

13

Objective #9: Can you factor trinomials in the form x2 + bx + c?

a) −2𝑥2 + 14𝑥 − 24 b) 𝑥2 − 9𝑥 + 8

c) y2 + 4y – 21 d) 𝑘3 + 5𝑘2 − 50𝑘

VOCABULARY: Difference of Two Perfect Squares

Factoring the differences of two squares

a2 – b2 = ( + ) ( – ) OR use the box method!

Example 2: Factor, if possible.

a) z2 – 81 b) −12𝑥6 + 27 c) 36a2 – 25b10

d) 2 – 50n8 e) b2 + 100 (be careful!)

Objective #10: Can you factor the difference of two squares?

a) −2𝑥2 + 98 b) −𝑥3 − 81𝑥 c) −4𝑎2 + 64

d) x2 - y2 e) 16x2 – 4y2

14

Steps for Solving Quadratic Equations by factoring:

1) Get a ________ on one side of the equation.

2) Factor

3) Use the zero-product property to find the answer(s), which are called ___________,

___________________, _____________________, or ____________________.

Example 3: Solve the following equations. Sketch a graph that includes the roots.

a) 2𝑥2 + 14𝑥 = 36 b) −𝑥2 + 144 = 0

c) 27𝑏2 = 72𝑏 d) −𝑥2 − 8𝑥 = 16

Objective #11: Can you solve quadratic equations and sketch a graph of the

solution?

a) 𝑥2 = 196 b) −𝑥2 − 𝑥 = −6 c) 2𝑥2 − 22𝑥 + 60 = 0

15

8.5 PRACTICE DAY

8.6 Guided Notes: Factoring and Solving Trinomials in the form ax2 + bx + c

Warm-Up

1) Simplify: 2) Simplify: 3) Simplify:

(2𝑥 + 3)(3𝑥2 − 4𝑥 + 7) (3𝑥2 − 4𝑥 + 12) + (2𝑥2 + 6𝑥 − 15) (2𝑥 − 1)(𝑥 − 5)

Factoring trinomials with a leading coefficient other than 1

Step 1: Make a chart to find two integers with a ________ = b and a _____________ = ac.

Step 2: Draw a 2x2 box.

Step 3: Put the terms in your box (_________ the middle term into 2 terms, look at your chart!)

Step 4: Factor.

Example 1: Factor 2x2 – llx + 5.

____ + ____ = -11

____ ∙ ____ = 10

Check:

Example 2: Factor each expression.

a) 3n2 + 2n - 8 b) 2y2 – 13y – 7

16

c) – 4x2 + 4x + 3

Objective #12: Can you factor trinomials in the form ax2 + bx + c?

a) 3x2 – 5x + 2 b) 2m2 + m – 21

c) 9y2 + 6y + 1 d) −𝑥2 − 10𝑥 − 25

17

Example 3: Solve the following quadratic equations and sketch a graph of the solutions.

a) 3𝑥2 − 𝑥 = 10 b) 6𝑥2 − 2𝑥 = 4

c) 0 = −4𝑥2 + 8𝑥 + 60 d) −3𝑥2 − 28𝑥 = −55

18

Objective #13: Can you factor and solve trinomials in the form ax2 + bx + c?

a) 0 = 2𝑥2 + 5𝑥 − 3 b) −4𝑥2 − 10𝑥 = −6

Reflect #6: Compare and contrast your answers and graphs for Objective 13.

8.7 Guided Notes: Factoring Completely Warm-Up

1) Factor: x2 + 5x + 6 2) Factor: 𝑥2 − 64

3) Simplify: (𝑥 + 7)2 4) Simplify: (4𝑥2 − 3𝑥 + 7) − (7𝑥2 − 6𝑥 + 2)

19

Trinomials with More Than One Variable (the Box Method)

Be careful when filling out the sections of your box…

Check your results with the original problem! Do you have all the variables?

Example 2: Factor each polynomial. a) x2 +14xy + 24y2 b) y2 – 10yz + 9z2

c) 9s2 + 6st + t2

Factoring Completely

Step 1: If possible, factor out a Greatest Common Factor.

Step 2: Can you factor the binomial or trinomial any further?

Step 3: Keep factoring until each portion of your answer is fully factored.

Example 3: Factor each polynomial, completely.

a) 5𝑎4– 405 b) 2x2 – 8x – 10

20

c) x4 – 16 d) −𝑥3 − 𝑥2 + 25𝑥 + 25

e) 3r3 – 21r2 + 30r f) 9d4 – 4d2

Objective #13: Can you factor completely?

a) 2y4 – 32 b) 49y2 – 25w6

c) a2 – 8ab + 16b2 d) y2 + 14yz + 49z2

21

Objective #14: Can you solve after factoring completely?

a) 0 = 45 – 80m2 b) –5r2 – 20r = 20

d) - x3 = 2x2 – 15x e) 4g2 + 20g + 24 = 0