Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor

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: Polynomials are a series of terms: 5x 3 + 4x 2 – 3x + 7 Term #1 Term #2Term #3 Term #4

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 :5x 2 y 3 Exponent: 2 Exponent: = 5

What we already know: A polynomial has a degree Degree of Polynomial: The degree of the highest term. Example :x 3 y - 5x 2 y 4 + 2xy Which degree is the largest? 6

Let’s Think... What objects have the characteristic of the #1? A unicycle has ONE wheel A mailbox with ONE flag

Let’s Think... What objects have the characteristic of the #2? A bicycle has TWO wheels A person has TWO eyes

Let’s Think... What objects have the characteristic of the #3? A tricycle has THREE wheels A clock has THREE hands

Wow!! Just like those objects, polynomials have the same characteristics!

Relate these polynomials to our objects we just discussed 2x 4x 3 + 3x - 1 3x 2 – 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…..

Monomial A unicycle has ONE wheel. This characteristic applies to a monomial. Monomial: A polynomial that has exactly ONE term

Binomial A bicycle has two wheels. What do you think this means for a binomial? Binomial: A polynomial that has exactly TWO terms

Trinomial A tricycle has three wheels. What do you think this means for a trinomial? Trinomial: A polynomial that has exactly THREE terms

Let’s Practice

Which of these are monomials? 3x 2 4x – 5 8x 2 + 2x – 1 5x + 7xx 2 – 4 x 3 + 2x x 5 2x 2 - 4

Which of these are monomials? 3x 2 5x + 7x = 12X 7 9x 5 Combine like terms!

Which of these are binomials? 3x 2 4x – 5 8x 2 + 2x – 1 5x + 7xx 2 – 4 x 3 + 2x x 5 2x 2 - 4

Which of these are binomials? 4x – 5 x 2 – 4 2x 2 - 4

Which of these are trinomials? 3x 2 4x – 5 8x 2 + 2x – 1 5x + 7xx 2 – 4 x 3 + 2x x 5 2x 2 - 4

Which of these are trinomials? 8x 2 + 2x – 1 x 3 + 2x x 3 + 2x + 11

Greatest Common Factor What do you think this means?

Definition Greatest Common Factor: the largest monomial that divides (is a factor of) each term of the polynomial. Often abbreviated as: GCF

Find 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… Use the 5 step method to find the greatest common factor of the following polynomial: 3x 3 + 6x 2 – 12x

1. What do we know? Trinomial Variables: x 3, x 2, and x Coefficients: 3, 6, and -12

3x 3 + 6x 2 – 12x 2.What must we find? Largest number that evenly divides each coefficient 3 Largest variable that evenly divides each x term. X

3x 3 + 6x 2 – 12x 3. Calculate GCF: 3*X=3X Largest number that divides each term

3x 3 + 6x 2 – 12x 3. Calculate GCF: 3*X=3X Largest variable that divides each term

3x 3 + 6x 2 – 12x 3. Calculate GCF: 3*X=3X GCF

3x 3 + 6x 2 – 12x 4. Rewrite our polynomial 3x(x 2 +2x – 4) GCF How did we get this?

3x(x 2 +2x – 4) Divide our GCF into each term of the polynomial. 3x 3 / 3x = x 2 6x 2 / 3x = 2x -12x / 3x = -4 Resulting answers are put inside the parenthesis!

3x 3 + 6x 2 – 12x 5. Check our answer. Multiply GCF through parenthesis: 3x(x 2 +2x – 4) = 3x 3 + 6x 2 – 12x The answers match, so our GCF is correct!

Polynomials Monomials One Term 3x Binomials Two Terms 3x Trinomials Three terms 5x 2 + 7x - 3

Summary Summary Monomial – Polynomial with Binomial – Polynomial with Trinomial – Polynomial with one term two terms three terms GCF stands for: Greatest Common Factor

Summary GCF: the largestmonomialthat divides evenly into eachterm of a polynomial.

Groups Group 1: Monomials Group 2: Binomials Group 3: Trinomials Group 4: Greatest Common Factor

Groups Receive poster board and markers -On poster board: 1.Write name of polynomial 2.Write definition of polynomial 3.Give 2 examples of polynomial 4.Draw picture to represent your polynomial.

Group 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.

Homework Complete worksheet: Due April 5 th