Warm Up
Multiply the polynomials
• 1. (𝑥 3 + 3𝑥 − 4)(𝑥 2 − 2𝑥 + 3)
Algebra 3
Chapter 6
Lesson 4
Factoring and Solving
Polynomial Equations
VOCAB
• Factor by grouping – is pairs of terms that
have common factors
• Quadratic form – is an expression in the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
CUBES
Number
Cubed
1
2
3
4
5
6
7
8
9
10
1
8
27
64
125
216
343
512
729
1000
Keep in Mind
• I’m trying to make this as easy as possible…
• I’m breaking this lesson into 3 easy parts
– It could’ve been 4 but I believe we can do the
zeros part already (we’ve done it in the past)
Warm Up
Factor the polynomial and find the zeros
• 1. 𝑥 3 + 64
• 2. 𝑥 3 + 216
Algebra 3
Chapter 6
Lesson 4
Factoring and Solving
Polynomial Equations
Special Cases
• Sum of Cubes
• 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )
• Difference of Cubes
• 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 )
Directions (Specials)
•
•
•
•
•
Figure out what a and b are
USE the awesome formula
Reduce
Set equal to zero (to find zeros)
Solve
I DO (Specials)
Factor and Find the zeros
• 1. 𝑥 3 + 27
• 2. 16𝑥 5 + 250𝑥 2
• 3. 𝑥 3 − 8
• 4. 1000𝑥 3 + 27
WE DO (Specials)
Factor and Find the zeros
• 1. 𝑥 3 + 64
• 2. 216𝑥 3 + 1
• 3. 𝑥 4 − 8𝑥
• 4. 32𝑥 3 − 4
YOU DO (Specials)
Factor and Find the zeros
• 1. 2𝑥 3 + 54
• 2. 27𝑥 7 − 216𝑥 4
• 3. 𝑥 3 + 729
• 4. 16𝑥 3 − 4
Question
• How do you know its going to be a sum or
difference of cubes?
Review
• Today you learned how to factor a cube
Homework
• Worksheet
– 6.4B (1 – 12)
Warm Up
Factor and find the zeros
• 1. (𝑥 3 − 729)
Algebra 3
Chapter 6
Lesson 4
Factoring and Solving
Polynomial Equations
Factor By Grouping
• Look at the first 2 terms
– Factor it
• Set equal to zero
• Solve
I DO (Grouping)
Factor and find zeros
• 1. 𝑥 3 + 𝑥 2 + 𝑥 + 1
• 2. 10𝑥 3 + 20𝑥 2 + 𝑥 + 2
• 3. 𝑥 3 + 3𝑥 2 + 10𝑥 + 30
• 4. 𝑥 3 − 2𝑥 2 + 4𝑥 − 8
WE DO (Grouping)
Factor and find zeros
• 1. 2𝑥 3 − 5𝑥 2 + 18𝑥 − 45
• 2. −2𝑥 3 − 4𝑥 2 − 3𝑥 − 6
• 3. 3𝑥 3 − 6𝑥 2 + 𝑥 − 2
• 4. 2𝑥 3 − 𝑥 2 + 2𝑥 − 1
YOU DO (Grouping)
Factor and find zeros
• 1. 3𝑥 3 − 2𝑥 2 − 9𝑥 + 6
• 2. 4𝑥 3 + 16𝑥 2 + 𝑥 + 4
• 3. 2𝑥 3 − 3𝑥 2 − 10𝑥 + 15
• 4. 𝑥 3 − 2𝑥 2 − 9𝑥 + 18
Review
• Today you learned how to factor polynomials
by grouping
Homework
• 6.4B (13 – 24)
Warm Up
Factor and find the zeros
• 1. 𝑥 3 + 3𝑥 2 + 2𝑥 + 6
Algebra 3
Chapter 6
Lesson 4
Factoring and Solving
Polynomial Equations
Factor By Grouping
• Look at the first 2 terms
– Factor it
• Set equal to zero
• Solve
I DO (Grouping)
Factor and find zeros
• 1.𝑥 3 − 𝑥 2 + 4𝑥 − 4
WE DO (Grouping)
Factor and find zeros
• 1.𝑥 3 + 5𝑥 2 + 𝑥 + 5
YOU DO (Grouping)
Factor and find zeros
• 1. 𝑥 3 + 4𝑥 2 + 3𝑥 + 12
Review
• Today you learned how to factor polynomials
by grouping
Homework
• 6.4B (13 – 24)
Warm Up
Factor and Find the Zeros
• 1. 3𝑥 4 + 6𝑥 3 − 24𝑥 − 48
Divide without a calculator
• 3425 by 15
Algebra 3
Chapter 6
Lesson 5
The Remainder and Factor
Theorems
VOCAB
• Polynomial Long Division – is one way to divide
polynomials
• Synthetic Division – Same thing as synthetic
substitution except with a big polynomial you take
the opposite of the constant term
• Remainder Theorem – if the polynomial, 𝑓(𝑥), is
divided by x-k then the remainder is 𝑟 = 𝑓(𝑘)
• Factor Theorem – a polynomial, 𝑓(𝑥), has a factor
x-k if and only if 𝑓 𝑘 = 0
• Divisor – the bottom number of a fraction or the
number you are dividing by
Directions (Long Division)
• THIS IS EXACTLY LIKE LONG DIVISION WITHOUT A
CALCULATOR
• Write the problem in standard form including the
missing terms
• Look at the first term in the divisor
– Find how many times that goes into the first term of the
polynomial
•
•
•
•
Multiply that answer times the WHOLE divisor
Subtract that from the polynomial
Keep going until you can’t do anymore
Remainder is then written over the divisor
I DO (Long Division)
Divide the polynomial
• 1. 2𝑥 4 + 3𝑥 3 + 5𝑥 − 1 by 𝑥 2 − 2𝑥 + 2
• 2. 𝑥 3 − 3𝑥 2 + 2𝑥 − 6 by 𝑥 2 + 3𝑥 − 1
• 3. 4𝑥 3 − 2𝑥 2 + 6𝑥 − 1 by 2𝑥 + 3
• 4. 2𝑥 4 + 3𝑥 − 1 by 𝑥 2 + 2𝑥 + 1
Warm Up
Divide using long division
• 1. −3𝑥 3 + 4𝑥 − 1 by 𝑥 − 3
Algebra 3
Chapter 6
Lesson 5
The Remainder and Factor
Theorems
Directions (Long Division)
• THIS IS EXACTLY LIKE LONG DIVISION WITHOUT A
CALCULATOR
• Write the problem in standard form including the
missing terms
• Look at the first term in the divisor
– Find how many times that goes into the first term of the
polynomial
•
•
•
•
Multiply that answer times the WHOLE divisor
Subtract that from the polynomial
Keep going until you can’t do anymore
Remainder is then written over the divisor
We Do (Long Division)
Divide the polynomial
• 1. 4𝑥 4 − 6𝑥 3 + 5 by 𝑥 2 − 4
• 2. 6𝑥 3 − 2𝑥 2 + 5 by 3𝑥 2 + 𝑥
• 3. 𝑥 3 − 2𝑥 2 + 5𝑥 − 1 by 3𝑥 + 2
• 4. 𝑥 3 + 5 by 2𝑥 2 − 1
You Do (Long Division)
Divide the polynomial
• 1. 2𝑥 3 − 4𝑥 2 + 3𝑥 + 5 by 4𝑥 2 + 2𝑥 − 1
• 2. 𝑥 2 + 3𝑥 − 6 by 𝑥 + 1
• 3. 2𝑥 3 − 7𝑥 2 − 17𝑥 − 3 by 2𝑥 + 3
Review
• What did you learn to do today?
Homework
• Worksheet
– 6.5B (1 – 8)
Warm Up
Divide the polynomials
• 1. 𝑥 3 + 5𝑥 2 − 2 by 𝑥 + 4
Algebra 3
Chapter 6
Lesson 5
The Remainder and Factor
Theorems
Question
• Does anyone know when you can use
synthetic division?
– Hint look at vocab
Directions (Synthetic Division)
• THIS IS EXACTLY LIKE SYNTHETIC SUBSTITUTION
• Write the problem in standard form including the
missing terms
• Only Write down the coefficients inside
• For the outside
– Remember to take the opposite of the number
• ANSWER
– YOU start with 1 power less than the original
I DO (Synthetic Division)
Divide the polynomial
• 1. 𝑥 3 − 2𝑥 2 − 6𝑥 − 9 by 𝑥 − 2
• 2. 𝑥 3 − 2𝑥 2 − 6𝑥 − 9 by 𝑥 + 3
• 3. 𝑥 3 − 8𝑥 + 3 by 𝑥 + 3
• 4. 𝑥 2 + 2𝑥 + 15 by 𝑥 − 3
We Do (Synthetic Division)
Divide the polynomial
• 1. 𝑥 4 − 16𝑥 2 + 𝑥 + 4 by x + 4
• 2. 𝑥 2 + 7𝑥 − 2 by 𝑥 − 2
• 3. 5𝑥 4 − 2𝑥 3 + 7𝑥 2 + 6𝑥 − 8 by 𝑥 − 2
• 4. 6𝑥 3 − 2𝑥 2 + 5𝑥 + 3 by 𝑥 + 3
You Do (Synthetic Division)
Divide the polynomial
• 1. 2𝑥 3 − 3𝑥 + 4 by 𝑥 − 1
• 2. 4𝑥 3 − 2𝑥 2 + 1 by 𝑥 + 2
• 3. 3𝑥 5 + 2𝑥 3 − 5𝑥 + 1 by 𝑥 − 3
• 4. 2𝑥 3 + 4𝑥 + 7𝑥 2 − 1 by 𝑥 + 1
Review
• Today you learned how to divide polynomials
using synthetic substitution
Homework
• Worksheet
– 6.5B (9 - 16)
Warm Up
IS it LONG DIVISION OR SYNTHETIC DIVISION
•
•
•
•
1. 4𝑥 3 − 2𝑥 2 + 1 by 𝑥 + 2
2. 3𝑥 2 − 1 by 𝑥 + 2
3. 𝑥 2 + 3𝑥 − 6 by 𝑥 2 + 1
4. 2𝑥 3 − 7𝑥 2 − 17𝑥 − 3 by 2𝑥 + 3
Algebra 3
Chapter 6
Lesson 5
The Remainder and Factor
Theorems
Directions (Finding Zero’s)
• Do synthetic Division of the function using the
given zero
• Set the answer equal to zero
• Solve
I DO (Finding Zero’s)
Find the zero’s of the polynomial by using the given
zero
• 1. 𝑥 3 − 2𝑥 2 − 9𝑥 + 18 ; 2
• 2. 𝑥 3 − 8𝑥 2 + 4𝑥 + 48 ; 4
• 3. 2𝑥 3 − 14𝑥 2 − 56𝑥 − 40 ; 10
• 4. 9𝑥 3 + 10𝑥 2 − 17𝑥 − 2 ; -2
WE DO (Finding Zero’s)
Find the zero’s of the polynomial by using the given
zero
• 1. 2𝑥 3 + 3𝑥 2 − 39𝑥 − 20 ; 4
• 2. 𝑥 3 − 14𝑥 2 + 47𝑥 − 18 ; 9
• 3. 𝑥 3 + 𝑥 2 + 2𝑥 + 24 ; -3
• 4. 𝑥 3 + 11𝑥 2 − 150𝑥 − 1512 ; -14
YOU DO (Finding Zero’s)
Find the zero’s of the polynomial by using the given
zero
• 1. 15𝑥 3 − 119𝑥 2 − 10𝑥 + 16 ; 8
• 2. 4𝑥 3 + 9𝑥 2 − 52𝑥 + 15 ; -5
• 3. 5𝑥 3 − 27𝑥 2 − 17𝑥 − 6 ; 6
• 4. 𝑥 3 + 𝑥 2 − 13𝑥 + 3 ; 3
Review
• Today what did you learn?
Homework
• Worksheet
– 6.5B (17 – 20)