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Chapter 10 Section 3 Identifying Polynomials Greatest Common Factor
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What we already know: Polynomial: a mathematical expression consisting of a sum of terms with each term including variables and constants
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What we already know: Polynomials are a series of terms: 5x 3 + 4x 2 – 3x + 7 Term #1 Term #2Term #3 Term #4
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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: 3 2 + 3 = 5
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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 +1 4620 Which degree is the largest? 6
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Let’s Think... What objects have the characteristic of the #1? A unicycle has ONE wheel A mailbox with ONE flag
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Let’s Think... What objects have the characteristic of the #2? A bicycle has TWO wheels A person has TWO eyes
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Let’s Think... What objects have the characteristic of the #3? A tricycle has THREE wheels A clock has THREE hands
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Wow!! Just like those objects, polynomials have the same characteristics!
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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…..
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Monomial A unicycle has ONE wheel. This characteristic applies to a monomial. Monomial: A polynomial that has exactly ONE term
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Binomial A bicycle has two wheels. What do you think this means for a binomial? Binomial: A polynomial that has exactly TWO terms
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Trinomial A tricycle has three wheels. What do you think this means for a trinomial? Trinomial: A polynomial that has exactly THREE terms
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Let’s Practice
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Which of these are monomials? 3x 2 4x – 5 8x 2 + 2x – 1 5x + 7xx 2 – 4 x 3 + 2x + 5 + 6 7 9x 5 2x 2 - 4
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Which of these are monomials? 3x 2 5x + 7x = 12X 7 9x 5 Combine like terms!
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Which of these are binomials? 3x 2 4x – 5 8x 2 + 2x – 1 5x + 7xx 2 – 4 x 3 + 2x + 5 + 6 7 9x 5 2x 2 - 4
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Which of these are binomials? 4x – 5 x 2 – 4 2x 2 - 4
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Which of these are trinomials? 3x 2 4x – 5 8x 2 + 2x – 1 5x + 7xx 2 – 4 x 3 + 2x + 5 + 6 7 9x 5 2x 2 - 4
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Which of these are trinomials? 8x 2 + 2x – 1 x 3 + 2x + 5 + 6 x 3 + 2x + 11
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Greatest Common Factor What do you think this means?
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Definition Greatest Common Factor: the largest monomial that divides (is a factor of) each term of the polynomial. Often abbreviated as: GCF
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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?
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2. What must we find? Largest number that divides into each coefficient (Factor tree) Largest variable that divides into each coefficient (smallest exponent)
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3. Calculate the GCF by multiplying the constant and variable you found in step #2.
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4. Rewrite our polynomial with the GCF.
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5. Check our answer!
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Step by Step… Use the 5 step method to find the greatest common factor of the following polynomial: 3x 3 + 6x 2 – 12x
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1. What do we know? Trinomial Variables: x 3, x 2, and x Coefficients: 3, 6, and -12
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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
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3x 3 + 6x 2 – 12x 3. Calculate GCF: 3*X=3X Largest number that divides each term
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3x 3 + 6x 2 – 12x 3. Calculate GCF: 3*X=3X Largest variable that divides each term
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3x 3 + 6x 2 – 12x 3. Calculate GCF: 3*X=3X GCF
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3x 3 + 6x 2 – 12x 4. Rewrite our polynomial 3x(x 2 +2x – 4) GCF How did we get this?
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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!
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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!
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Polynomials Monomials One Term 3x Binomials Two Terms 3x 2 - 7 Trinomials Three terms 5x 2 + 7x - 3
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Summary Summary Monomial – Polynomial with Binomial – Polynomial with Trinomial – Polynomial with one term two terms three terms GCF stands for: Greatest Common Factor
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Summary GCF: the largestmonomialthat divides evenly into eachterm of a polynomial.
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Groups Group 1: Monomials Group 2: Binomials Group 3: Trinomials Group 4: Greatest Common Factor
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Groups 1 - 3 -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.
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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.
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Homework Complete worksheet: Due April 5 th
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