unit 8 notes: solving quadratics by factoring alg...
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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
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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.
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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)
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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)
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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)
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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
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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)
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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
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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
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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.
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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) ?
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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