Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor.
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Transcript of Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor.
Chapter 10 Chapter 10 Section 3Section 3
Identifying PolynomialsIdentifying Polynomials
Greatest Common FactorGreatest Common Factor
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
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
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
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
Let’s Think. . . Let’s Think. . . What objects have the characteristic of
the #1?
A unicycle has ONE wheel
A mailbox with ONE flag
Let’s Think. . . Let’s Think. . . What objects have the characteristic of
the #2?
A bicycle has TWO wheels
A person has TWO eyes
Let’s Think. . . Let’s Think. . . What objects have the characteristic of
the #3?
A tricycle has THREE wheels
A clock has THREE hands
Wow!!Wow!!
Just like those objects, polynomials have the same characteristics!
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…..
MonomialMonomial
A unicycle has ONE wheel. This characteristic applies to a monomial.
Monomial: A polynomial that has exactly ONE term
BinomialBinomial
A bicycle has two wheels. What do you think this means for a binomial?
Binomial: A polynomial that has exactly TWO terms
TrinomialTrinomial
A tricycle has three wheels. What do you think this means for a trinomial?
Trinomial: A polynomial that has exactly THREE terms
Let’s PracticeLet’s Practice
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
Which of these are monomials?Which of these are monomials?
3x2
5x + 7x = 12X
7 9x5
Combine like terms!
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
Which of these are binomials?Which of these are binomials?
4x – 5
x2 – 4
2x2 - 4
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
Which of these are trinomials?Which of these are trinomials?
8x2 + 2x – 1
x3 + 2x + 5 + 6
x3 + 2x + 11
Greatest Common FactorGreatest Common Factor
What do you think this means?
DefinitionDefinition
Greatest Common Factor: the largest monomial that divides (is a factor of) each term of the
polynomial.
Often abbreviated as: GCF
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?
2. What must we find?
• Largest number that divides into each coefficient (Factor tree)
• Largest variable that divides into each coefficient (smallest exponent)
3. Calculate the GCF by multiplying the constant and variable you found in step #2.
4. Rewrite our polynomial with the GCF.
5. Check our answer!
Step by Step…Step by Step…
Use the 5 step method to find the greatest common factor of the following polynomial:
3x3 + 6x2 – 12x
3x3x33 + 6x + 6x22 – 12x – 12x 1. What do we know?
Trinomial
Variables: x3, x2, and x
Coefficients: 3, 6, and -12
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
3x3x33 + 6x + 6x22 – 12x – 12x 3. Calculate GCF:
3 * X = 3X
Largest number that divides each term
3x3x33 + 6x + 6x22 – 12x – 12x 3. Calculate GCF:
3 * X = 3X
Largest variable that divides each term
3x3x33 + 6x + 6x22 – 12x – 12x 3. Calculate GCF:
3 * X = 3X
GCF
3x3x33 + 6x + 6x22 – 12x – 12x 4. Rewrite our polynomial
3x(x2 +2x – 4)
GCF
How did we get this?
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!
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!
Polynomials
Monomials Binomials Trinomials
One Term
3x
Two Terms
3x2 - 7
Three terms
5x2 + 7x - 3
SummarySummary
Monomial – Polynomial with
Binomial – Polynomial with
Trinomial – Polynomial with
one term
two terms
three terms
GCF stands for: Greatest Common Factor
SummarySummaryGCF: the largest monomial that
divides evenly into each
term
of a polynomial.
GroupsGroups
Group 1: MonomialsGroup 2: Binomials
Group 3: TrinomialsGroup 4: Greatest Common Factor
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.
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.
HomeworkHomework
Complete worksheet:
Due April 5th