chapter 10 section 3 identifying polynomials greatest common factor
TRANSCRIPT
![Page 1: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/1.jpg)
Chapter 10 Chapter 10 Section 3Section 3
Identifying PolynomialsIdentifying Polynomials
Greatest Common FactorGreatest Common Factor
![Page 2: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/2.jpg)
What we already know:What we already know:
Polynomial: a mathematical expression consisting of a sum of terms with each term including variables and constants
![Page 3: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/3.jpg)
What we already know:What we already know:
Polynomials are a series of terms:
5x3 + 4x2 – 3x + 7
Term #1 Term #2 Term #3Term #4
![Page 4: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/4.jpg)
What we already know:What we already know:
Each term in a polynomial has a “degree”
Degree of Term: The sum of the individual exponents in the term.
Example :5x2y3
Exponent: 2 Exponent: 3
2 + 3 = 5
![Page 5: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/5.jpg)
What we already know:What we already know:
A polynomial has a degree
Degree of Polynomial: The degree of the highest term.
Example :x3y - 5x2y4 + 2xy +1
4 6 2 0
Which degree is the largest?
6
![Page 6: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/6.jpg)
Let’s Think. . . Let’s Think. . . What objects have the characteristic of
the #1?
A unicycle has ONE wheel
A mailbox with ONE flag
![Page 7: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/7.jpg)
Let’s Think. . . Let’s Think. . . What objects have the characteristic of
the #2?
A bicycle has TWO wheels
A person has TWO eyes
![Page 8: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/8.jpg)
Let’s Think. . . Let’s Think. . . What objects have the characteristic of
the #3?
A tricycle has THREE wheels
A clock has THREE hands
![Page 9: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/9.jpg)
Wow!!Wow!!
Just like those objects, polynomials have the same characteristics!
![Page 10: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/10.jpg)
Relate these polynomials to Relate these polynomials to our objects we just discussedour objects we just discussed
2x
4x3 + 3x - 1
3x2 – 4
Unicycle with one wheel…...
Polynomial with one term…..
Person with two eyes…...
Polynomial with two terms…..
Clock with three hands…...
Polynomial with three terms…..
![Page 11: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/11.jpg)
MonomialMonomial
A unicycle has ONE wheel. This characteristic applies to a monomial.
Monomial: A polynomial that has exactly ONE term
![Page 12: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/12.jpg)
BinomialBinomial
A bicycle has two wheels. What do you think this means for a binomial?
Binomial: A polynomial that has exactly TWO terms
![Page 13: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/13.jpg)
TrinomialTrinomial
A tricycle has three wheels. What do you think this means for a trinomial?
Trinomial: A polynomial that has exactly THREE terms
![Page 14: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/14.jpg)
Let’s PracticeLet’s Practice
![Page 15: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/15.jpg)
Which of these are monomials?Which of these are monomials?
3x2 4x – 5 8x2 + 2x – 1
5x + 7x x2 – 4 x3 + 2x + 5 + 6
7 9x5 2x2 - 4
![Page 16: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/16.jpg)
Which of these are monomials?Which of these are monomials?
3x2
5x + 7x = 12X
7 9x5
Combine like terms!
![Page 17: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/17.jpg)
Which of these are binomials?Which of these are binomials?
3x2 4x – 5 8x2 + 2x – 1
5x + 7x x2 – 4 x3 + 2x + 5 + 6
7 9x5 2x2 - 4
![Page 18: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/18.jpg)
Which of these are binomials?Which of these are binomials?
4x – 5
x2 – 4
2x2 - 4
![Page 19: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/19.jpg)
Which of these are trinomials?Which of these are trinomials?
3x2 4x – 5 8x2 + 2x – 1
5x + 7x x2 – 4 x3 + 2x + 5 + 6
7 9x5 2x2 - 4
![Page 20: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/20.jpg)
Which of these are trinomials?Which of these are trinomials?
8x2 + 2x – 1
x3 + 2x + 5 + 6
x3 + 2x + 11
![Page 21: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/21.jpg)
Greatest Common FactorGreatest Common Factor
What do you think this means?
![Page 22: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/22.jpg)
DefinitionDefinition
Greatest Common Factor: the largest monomial that divides (is a factor of) each term of the
polynomial.
Often abbreviated as: GCF
![Page 23: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/23.jpg)
Find GCFFind GCF
To find the GCF, there are 5 steps to follow:
1. What do we know about the polynomial?
• How many terms?
• Monomial, Binomial or Trinomial?
![Page 24: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/24.jpg)
2. What must we find?
• Largest number that divides into each coefficient (Factor tree)
• Largest variable that divides into each coefficient (smallest exponent)
![Page 25: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/25.jpg)
3. Calculate the GCF by multiplying the constant and variable you found in step #2.
![Page 26: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/26.jpg)
4. Rewrite our polynomial with the GCF.
![Page 27: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/27.jpg)
5. Check our answer!
![Page 28: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/28.jpg)
Step by Step…Step by Step…
Use the 5 step method to find the greatest common factor of the following polynomial:
3x3 + 6x2 – 12x
![Page 29: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/29.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x 1. What do we know?
Trinomial
Variables: x3, x2, and x
Coefficients: 3, 6, and -12
![Page 30: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/30.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x 2. What must we find?
Largest number that evenly divides each coefficient 3
Largest variable that evenly divides each x term. X
![Page 31: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/31.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x 3. Calculate GCF:
3 * X = 3X
Largest number that divides each term
![Page 32: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/32.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x 3. Calculate GCF:
3 * X = 3X
Largest variable that divides each term
![Page 33: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/33.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x 3. Calculate GCF:
3 * X = 3X
GCF
![Page 34: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/34.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x 4. Rewrite our polynomial
3x(x2 +2x – 4)
GCF
How did we get this?
![Page 35: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/35.jpg)
3x(x2 +2x – 4)Divide our GCF into
each term of the polynomial.
3x3 / 3x = x2
6x2 / 3x = 2x
-12x / 3x = -4
Resulting answers are put inside the parenthesis!
![Page 36: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/36.jpg)
3x3x33 + 6x + 6x22 – 12x – 12x5. Check our answer.
Multiply GCF through parenthesis:
3x(x2 +2x – 4) = 3x3 + 6x2 – 12x
The answers match, so our GCF is correct!
![Page 37: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/37.jpg)
Polynomials
Monomials Binomials Trinomials
One Term
3x
Two Terms
3x2 - 7
Three terms
5x2 + 7x - 3
![Page 38: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/38.jpg)
SummarySummary
Monomial – Polynomial with
Binomial – Polynomial with
Trinomial – Polynomial with
one term
two terms
three terms
GCF stands for: Greatest Common Factor
![Page 39: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/39.jpg)
SummarySummaryGCF: the largest monomial that
divides evenly into each
term
of a polynomial.
![Page 40: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/40.jpg)
GroupsGroups
Group 1: MonomialsGroup 2: Binomials
Group 3: TrinomialsGroup 4: Greatest Common Factor
![Page 41: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/41.jpg)
Groups 1 - 3Groups 1 - 3- Receive poster board and markers
- On poster board:
1. Write name of polynomial 2. Write definition of polynomial3. Give 2 examples of polynomial4. Draw picture to represent your
polynomial.
![Page 42: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/42.jpg)
Group 4Group 4- Receive poster board and markers
- On poster board:
- Write Greatest Common Factor- Write definition of GCF- Write 5 steps to find the GCF- Develop a clever way of
remembering the 5 steps.
![Page 43: Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor](https://reader035.vdocuments.us/reader035/viewer/2022062221/56649ef15503460f94c02622/html5/thumbnails/43.jpg)
HomeworkHomework
Complete worksheet:
Due April 5th