Warm Up Introduction to Polynomials and Adding and Subtracting Polynomials.

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Presentation transcript:

Warm Up Introduction to Polynomials and Adding and Subtracting Polynomials

Classifying Polynomials: We classify polynomials based on the number of _________. What is a monomial? What is a binomial? What is a trinomial? When do we use the term polynomial? terms A polynomial with only 1 term. A polynomial with 2 terms. A polynomial with 3 terms. We use the term polynomial to describe any expression which contains some combination of variables and numbers.

What is a term? Terms are made up of numbers, variables or the product and/or quotient of some combination of numbers and variables. 2x EXAMPLE: 2x is a term. (the product of a # and variable) 2x/3 2x/3 is a term. (the quotient of a # and variable) Terms are separated by ________ and ________ signs! + -

Classify each of the polynomials 1.3x 2 y + 3x + 4y 2.2x 2 y 4 z 3.2xy 2 + 3x 2 y + 9x 4.5x 2 y + 2xy 3 Trinomial Monomial Trinomial Binomial

Calculating the degree of a polynomial.. If it is a single variable term, find the term with the largest exponent. 5x 3 + 9x 2 + x This is a 3rd degree polynomial. Remember the understood 1!!! 1 If it is a multi-variable term, you add the exponents of all of the variables. 4x 2 y + 2xy This is a 4th degree polynomial (3+1=4). Don’t forget the understood 1’s!!!! 11

Calculate the degree of the polynomial… 1.6x 2 + 4x + 4x 3 + x 2.21x 2 y + 12x 4 y 2 + 2xy 4 3.3x 2 + 2y 3 3rd degree 6th degree 3rd degree

Writing a polynomial in standard form… Standard form is in descending order of the x variable. 1.2xy 3 + 3x 2 y + x 3 2.3xy 2 + 3x 3 + 2x 2 y X 3 + 3x 2 y + 2xy 3 3x 3 + 2x 2 y + 3xy 2

Adding and Subtracting Polynomials… is nothing more than combining like terms. Remember you can do this vertically or horizontally. 1.(2x 2 - 3x + 3) + (4x 2 + 5x - 9) 2.(3x 2 - 9x - 5) - (-2x 2 - 4x + 5) 6x 2 + 2x -6 5x 2 - 5x (x 2 - x - 2)

Ex 1 (x 2 + 3x + 4) + (-2x x - 5) Combine LIKE terms x 2 + 3x x x x x - 1 Final Answer

Ex 2. (4b 3 - 2b) + (b 3 + 6b 2 + 3b - 7) 4b 3 + 0b 2 - 2b + 0 b 3 + 6b 2 + 3b - 7 5b 3 + 6b 2 + 1b - 7 Final Answer

Ex 3. (12y 2 - 8y + 4) - (9y 2 + 5y + 1) Don’t forget to Distribute the -1!! (12y 2 - 8y + 4) - 1(9y 2 + 5y + 1) (12y 2 - 8y + 4) - 9y 2 - 5y y 2 - 8y y 2 - 5y - 1 3y y + 3 Final Answer

Ex 4. (3a a - 15) - (-a 3 + 2a 2 + 6a - 9) (3a a - 15) - 1(-a 3 + 2a 2 + 6a - 9) 3a a a 3 - 2a 2 - 6a + 9 3a 3 + 0a a a 3 - 2a 2 - 6a + 9 4a 3 - 2a 2 + 4a - 6 Final Answer

POLYNOMIAL MULTIPLICATION (2x - 3)(x + 5) 3 methods a.Distribute (2x - 3)(x + 5) *4 multiplications 2x x - 3x - 15 (then combine like terms) 2x x - 3x x 2 + 7x - 15 Final Answer

b. 3rd grade style 2x - 3 x x x 2 - 3x + 0 2x 2 + 7x - 15 Final Answer!

c. Box Method 2x - 3 x 2x 2 -3x x -15 2x 2 + 7x - 15 Final Answer

Ex 5. (4x + 7)(3x + 7) a.) 12x x + 21x x x + 49 b.) 4x + 7c.) 4x + 7 3x + 7 3x 12x 2 +21x +28x x x x x x + 49

Ex 6. (x - 1)(x 2 - 4x + 6) a.x 3 - 4x 2 + 6x - 1x 2 + 4x - 6 x 3 - 5x x - 6 b. x 2 - 4x + 6c. x 2 - 4x + 6 x - 1 x x 3 -4x 2 +6x -1x 2 + 4x x 2 +4x - 6 1x 3 -4x 2 + 6x + 0 x 3 - 5x x - 6 x 3 - 5x x - 6

Ex 7. (2z 2 + 3z - 4)(4z + 5) a.8z z z z - 16z z z 2 - 1z - 20 b. 2z 2 + 3z - 4 c. 2z 2 + 3z - 4 4z + 5 4z 8z 3 +12z 2 -16z +10z z z 2 +15z z 3 +12z z + 0 8z z 2 - 1z - 20

Ex 8. (2a + 5)(2a - 5) a.4a a + 10a a b. 2a + 5c.2a +5 2a - 5 2a 4a 2 +10a -10a a -25 4a 2 +10a + 0 4a

Ex 9. (3m + 4n) 2 a.(3m + 4n)(3m + 4n) 9m mn + 12mn + 16n 2 9m mn + 16n 2 b.3m + 4n c. 3m +4n 3m + 4n 3m 9m 2 +12mn +12mn + 16n 2 +4n +12mn +16n 2 9m mn + 0 9m mn + 16n 2