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